Continuous-time quantum walk-based ans\"atze on neutral atom hardware

TL;DR

This paper demonstrates the first experimental implementation of continuous-time quantum walk-based variational algorithms on neutral-atom hardware, showing super-quadratic convergence and efficient entangled state preparation, paving the way for practical quantum speedups.

Contribution

It introduces a novel approach to implement quantum walk-based ans"atze on neutral-atom processors with minimal calibration and demonstrates their effectiveness for both unentangled and entangled states.

Findings

First experimental realization of quantum walk-based ans"atze on neutral-atom hardware.

Observation of super-quadratic convergence at low circuit depth.

Efficient preparation of symmetric entangled states with spectral gap scaling.

Abstract

Continuous-time quantum walks offer provable speedups for certain computational problems, yet translating these advantages to near-term hardware remains challenging. We present the first experimental demonstration of variational ans\"atze based on continuous-time quantum walks on an analog neutral-atom processor. For unentangled targets, we derive closed-form expressions for near-optimal control parameters that transfer directly to hardware with minimal calibration. Experiments on QuEra's Aquila processor provide the first observation of the super-quadratic convergence characteristic of efficient quantum walk algorithms, visible at low circuit depth, with theory predicting stronger speedups as hardware improves. For entangled targets, specifically symmetric superpositions in the Rydberg-blockaded subspace, we introduce an optimization protocol exploiting spectral properties of the walk dynamics. The required evolution time scales inversely with the spectral gap, offering an advantage over adiabatic protocols that scale to the square of the spectral gap. We demonstrate this scaling behavior on Aquila and verify that the prepared states are coherent superpositions via quench dynamics. This constitutes the first preparation of such symmetric entangled states on neutral-atom hardware. Our results establish a practical pathway from abstract quantum walk algorithms to analog quantum processors, demonstrating that the dynamics underlying their potential for super-quadratic quantum speedup are accessible on current devices.

Figures

Tables

Table 1. Table 2: Numerical and hardware results for | z half ⟩ |z^{\text{half}}\rangle and | z MIS ⟩ |z^{\text{MIS}}\rangle product states, as detailed at the start of C . Highlighted rows emphasize the target state of Fig.  5

$N$	$\left|𝒱\right|$	State	Depth ($p$)	${\tau }_0$	${\tau }_1$	$J_{\mathrm{eff}}$	Naïve Count	Perfect	Emulation	EM (H)	EM (E)
$5$	$11$	$00101$	$1$	$0.532$	$1.146$	$0.936$	278/954	$0.964$	$0.481$	$0.32[0.28,0.35]$	$0.48[0.44,0.51]$
$5$	$11$	$00101$	$2$	$0.330$	$0.642$	$0.973$	166/948	$1.000$	$0.554$	$0.15[0.12,0.18]$	$0.56[0.52,0.59]$
$5$	$11$	$00101$	$3$	$0.225$	$0.456$	$0.987$	127/955	$1.000$	$0.503$	$0.08[0.05,0.10]$	$0.50[0.45,0.54]$
$6$	$18$	$000101$	$1$	$0.606$	$0.960$	$1.003$	444/1882	$0.789$	$0.396$	$0.28[0.25,0.30]$	$0.40[0.35,0.44]$
$6$	$18$	$000101$	$2$	$0.268$	$0.655$	$0.995$	456/1874	$0.915$	$0.504$	$0.26[0.24,0.29]$	$0.51[0.46,0.55]$
$6$	$18$	$000101$	$3$	$0.275$	$0.435$	$0.995$	314/1870	$0.963$	$0.620$	$0.15[0.12,0.17]$	$0.59[0.54,0.64]$
$7$	$29$	$0000101$	$1$	$0.639$	$0.878$	$1.035$	336/1884	$0.704$	$0.356$	$0.21[0.18,0.23]$	$0.38[0.34,0.43]$
$7$	$29$	$0000101$	$2$	$0.226$	$0.663$	$1.012$	632/1855	$0.857$	$0.613$	$0.47[0.43,0.50]$	$0.63[0.58,0.68]$
$7$	$29$	$0000101$	$3$	$0.302$	$0.423$	$0.999$	454/1862	$0.947$	$0.669$	$0.28[0.25,0.30]$	$0.69[0.64,0.73]$
$8$	$47$	$00010101$	$1$	$0.578$	$0.992$	$1.001$	351/1857	$0.706$	$0.487$	$0.23[0.21,0.26]$	$0.45[0.40,0.50]$
$8$	$47$	$00010101$	$2$	$0.285$	$0.649$	$0.992$	314/1859	$0.911$	$0.511$	$0.19[0.16,0.21]$	$0.51[0.46,0.55]$
$8$	$47$	$00010101$	$3$	$0.264$	$0.439$	$0.994$	255/1902	$0.957$	$0.614$	$0.12[0.10,0.15]$	$0.63[0.58,0.68]$
$9$	$76$	$000010101$	$1$	$0.615$	$0.908$	$1.032$	144/929	$0.657$	$0.374$	$0.18[0.15,0.22]$	$0.39[0.35,0.43]$
$9$	$76$	$000010101$	$2$	$0.250$	$0.656$	$1.006$	361/1841	$0.849$	$0.596$	$0.26[0.23,0.29]$	$0.60[0.55,0.65]$
$9$	$76$	$000010101$	$3$	$0.289$	$0.428$	$0.998$	202/1841	$0.929$	$0.551$	$0.10[0.08,0.12]$	$0.54[0.49,0.60]$
$10$	$123$	$0000010101$	$1$	$0.636$	$0.861$	$1.049$	276/1822	$0.620$	$0.397$	$0.18[0.16,0.20]$	$0.40[0.34,0.44]$
$10$	$123$	$0000010101$	$2$	$0.223$	$0.661$	$1.016$	238/944	$0.796$	$0.552$	$0.35[0.31,0.39]$	$0.54[0.49,0.59]$
$10$	$123$	$0000010101$	$3$	$0.305$	$0.422$	$1.000$	247/1844	$0.926$	$0.584$	$0.17[0.15,0.19]$	$0.61[0.55,0.66]$
$11$	$199$	$00000010101$	$1$	$0.650$	$0.831$	$1.061$	115/914	$0.564$	$0.326$	$0.14[0.11,0.17]$	$0.31[0.27,0.36]$
$11$	$199$	$00000010101$	$2$	$0.202$	$0.665$	$1.025$	333/1828	$0.748$	$0.474$	$0.27[0.24,0.29]$	$0.52[0.46,0.57]$
$11$	$199$	$00000010101$	$3$	$0.317$	$0.417$	$1.002$	135/1812	$0.913$	$0.442$	$0.08[0.06,0.10]$	$0.42[0.37,0.47]$
$12$	$322$	$000001010101$	$1$	$0.620$	$0.885$	$1.044$	172/1821	$0.525$	$0.298$	$0.13[0.11,0.15]$	$0.25[0.21,0.30]$
$12$	$322$	$000001010101$	$2$	$0.241$	$0.657$	$1.011$	252/1792	$0.761$	$0.514$	$0.21[0.18,0.24]$	$0.52[0.47,0.58]$
$12$	$322$	$000001010101$	$3$	$0.296$	$0.426$	$0.999$	123/1800	$0.919$	$0.512$	$0.07[0.05,0.09]$	$0.56[0.50,0.60]$
$13$	$521$	$0000001010101$	$1$	$0.635$	$0.852$	$1.056$	135/1765	$0.483$	$0.324$	$0.11[0.08,0.13]$	$0.32[0.27,0.37]$
$13$	$521$	$0000001010101$	$2$	$0.221$	$0.660$	$1.019$	253/1784	$0.755$	$0.510$	$0.23[0.20,0.26]$	$0.51[0.46,0.58]$
$13$	$521$	$0000001010101$	$3$	$0.308$	$0.421$	$1.001$	122/1802	$0.892$	$0.562$	$0.09[0.07,0.11]$	$0.54[0.48,0.59]$
$14$	$843$	$00000001010101$	$1$	$0.646$	$0.829$	$1.065$	132/1806	$0.482$	$0.288$	$0.09[0.07,0.11]$	$0.29[0.25,0.35]$
$14$	$843$	$00000001010101$	$2$	$0.205$	$0.664$	$1.025$	279/1804	$0.689$	$0.463$	$0.25[0.22,0.28]$	$0.43[0.38,0.50]$
$14$	$843$	$00000001010101$	$3$	$0.317$	$0.417$	$1.002$	120/1814	$0.882$	$0.500$	$0.09[0.07,0.11]$	$0.52[0.47,0.59]$
$15$	$1364$	$000000001010101$	$1$	$0.654$	$0.812$	$1.071$	121/1756	$0.459$	$0.296$	$0.10[0.08,0.12]$	$0.31[0.26,0.35]$
$15$	$1364$	$000000001010101$	$2$	$0.191$	$0.667$	$1.031$	245/1767	$0.684$	$0.485$	$0.27[0.24,0.29]$	$0.52[0.48,0.60]$
$15$	$1364$	$000000001010101$	$3$	$0.324$	$0.414$	$1.003$	126/1776	$0.884$	$0.544$	$0.11[0.09,0.13]$	$0.53[0.48,0.60]$
$16$	$2207$	$0000000101010101$	$1$	$0.635$	$0.847$	$1.060$	78/1754	$0.399$	$0.221$	$0.07[0.05,0.09]$	$0.27[0.23,0.32]$
$16$	$2207$	$0000000101010101$	$2$	$0.220$	$0.660$	$1.020$	153/1749	$0.664$	$0.439$	$0.15[0.12,0.17]$	$0.49[0.44,0.56]$
$16$	$2207$	$0000000101010101$	$3$	$0.309$	$0.420$	$1.001$	40/1727	$0.896$	$0.412$	$0.03[0.02,0.04]$	$0.42[0.37,0.49]$
$17$	$3571$	$00000000101010101$	$1$	$0.644$	$0.828$	$1.067$	74/1740	$0.385$	$0.206$	$0.06[0.04,0.08]$	$0.22[0.18,0.26]$
$17$	$3571$	$00000000101010101$	$2$	$0.207$	$0.663$	$1.025$	106/1700	$0.662$	$0.440$	$0.10[0.08,0.12]$	$0.42[0.37,0.49]$
$17$	$3571$	$00000000101010101$	$3$	$0.317$	$0.417$	$1.002$	60/1741	$0.856$	$0.446$	$0.06[0.04,0.08]$	$0.49[0.43,0.53]$
$18$	$5778$	$000000000101010101$	$1$	$0.651$	$0.814$	$1.072$	66/1706	$0.370$	$0.246$	$0.05[0.04,0.07]$	$0.24[0.20,0.28]$
$18$	$5778$	$000000000101010101$	$2$	$0.195$	$0.665$	$1.030$	166/1711	$0.637$	$0.419$	$0.22[0.19,0.24]$	$0.45[0.40,0.51]$
$18$	$5778$	$000000000101010101$	$3$	$0.323$	$0.414$	$1.003$	52/1685	$0.857$	$0.442$	$0.06[0.04,0.07]$	$0.39[0.34,0.45]$
$19$	$9349$	$0000000000101010101$	$1$	$0.657$	$0.802$	$1.076$	67/1713	$0.350$	$0.204$	$0.07[0.05,0.08]$	$0.21[0.17,0.26]$
$19$	$9349$	$0000000000101010101$	$2$	$0.184$	$0.667$	$1.034$	118/1704	$0.622$	$0.386$	$0.15[0.12,0.18]$	$0.47[0.41,0.52]$
$19$	$9349$	$0000000000101010101$	$3$	$0.328$	$0.412$	$1.004$	56/1678	$0.856$	$0.446$	$0.07[0.05,0.09]$	$0.48[0.41,0.51]$
$20$	$\mathrm{15\hspace{0.17em}127}$	$00000000010101010101$	$1$	$0.643$	$0.828$	$1.068$	40/1698	$0.352$	$0.161$	$0.03[0.02,0.05]$	$0.20[0.16,0.24]$
$20$	$\mathrm{15\hspace{0.17em}127}$	$00000000010101010101$	$2$	$0.208$	$0.662$	$1.025$	112/1753	$0.608$	$0.356$	$0.14[0.12,0.17]$	$0.35[0.31,0.44]$
$20$	$\mathrm{15\hspace{0.17em}127}$	$00000000010101010101$	$3$	$0.316$	$0.417$	$1.002$	23/1721	$0.837$	$0.331$	$0.02[0.01,0.04]$	$0.36[0.30,0.41]$
$21$	$\mathrm{24\hspace{0.17em}476}$	$000000000010101010101$	$1$	$0.649$	$0.815$	$1.073$	35/1708	$0.324$	$0.175$	$0.03[0.01,0.04]$	$0.19[0.14,0.22]$
$21$	$\mathrm{24\hspace{0.17em}476}$	$000000000010101010101$	$2$	$0.198$	$0.664$	$1.029$	58/1726	$0.558$	$0.319$	$0.07[0.05,0.09]$	$0.36[0.31,0.42]$
$21$	$\mathrm{24\hspace{0.17em}476}$	$000000000010101010101$	$3$	$0.322$	$0.415$	$1.003$	30/1722	$0.851$	$0.407$	$0.02[0.01,0.04]$	$0.41[0.35,0.46]$
$22$	$\mathrm{39\hspace{0.17em}603}$	$0000000000010101010101$	$1$	$0.655$	$0.804$	$1.077$	39/1696	$0.322$	$0.192$	$0.04[0.02,0.05]$	$0.21[0.16,0.24]$
$22$	$\mathrm{39\hspace{0.17em}603}$	$0000000000010101010101$	$2$	$0.189$	$0.666$	$1.033$	106/1719	$0.557$	$0.335$	$0.17[0.14,0.20]$	$0.44[0.38,0.49]$
$22$	$\mathrm{39\hspace{0.17em}603}$	$0000000000010101010101$	$3$	$0.326$	$0.413$	$1.004$	13/1691	$0.827$	$0.368$	$0.010[0.000,0.019]$	$0.43[0.37,0.48]$
$23$	$\mathrm{64\hspace{0.17em}079}$	$00000000000010101010101$	$1$	$0.659$	$0.796$	$1.080$	33/1652	$0.322$	$0.150$	$0.04[0.02,0.05]$	$0.17[0.13,0.20]$
$23$	$\mathrm{64\hspace{0.17em}079}$	$00000000000010101010101$	$2$	$0.180$	$0.668$	$1.036$	64/1631	$0.546$	$0.346$	$0.13[0.10,0.15]$	$0.47[0.39,0.49]$
$23$	$\mathrm{64\hspace{0.17em}079}$	$00000000000010101010101$	$3$	$0.330$	$0.411$	$1.004$	19/1667	$0.838$	$0.382$	$0.02[0.01,0.04]$	$0.41[0.35,0.46]$
$6$	$18$	$010101$	$1$	$0.439$	$1.202$	$0.957$	534/1880	$0.990$	$0.617$	$0.32[0.29,0.35]$	$0.60[0.56,0.64]$
$6$	$18$	$010101$	$2$	$0.340$	$0.632$	$0.980$	74/860	$0.998$	$0.232$	$0.06[0.03,0.08]$	$0.22[0.18,0.26]$
$7$	$29$	$0010101$	$1$	$0.506$	$1.153$	$0.947$	475/1889	$0.954$	$0.633$	$0.29[0.26,0.32]$	$0.63[0.59,0.66]$
$7$	$29$	$0010101$	$2$	$0.332$	$0.639$	$0.975$	157/860	$0.999$	$0.536$	$0.18[0.14,0.21]$	$0.52[0.47,0.56]$
$8$	$47$	$01010101$	$1$	$0.439$	$1.201$	$0.958$	353/1862	$0.981$	$0.690$	$0.22[0.19,0.25]$	$0.73[0.68,0.77]$
$8$	$47$	$01010101$	$2$	$0.340$	$0.632$	$0.980$	57/821	$0.998$	$0.364$	$0.07[0.05,0.10]$	$0.40[0.36,0.44]$
$9$	$76$	$001010101$	$1$	$0.492$	$1.160$	$0.950$	342/1821	$0.930$	$0.675$	$0.23[0.20,0.26]$	$0.68[0.64,0.72]$
$9$	$76$	$001010101$	$2$	$0.334$	$0.637$	$0.977$	115/840	$0.996$	$0.540$	$0.14[0.11,0.17]$	$0.53[0.48,0.57]$
$10$	$123$	$0101010101$	$1$	$0.439$	$1.201$	$0.958$	202/933	$0.991$	$0.722$	$0.29[0.25,0.33]$	$0.73[0.68,0.77]$
$10$	$123$	$0101010101$	$2$	$0.340$	$0.632$	$0.980$	54/809	$0.998$	$0.390$	$0.07[0.04,0.10]$	$0.40[0.35,0.44]$
$11$	$199$	$00101010101$	$1$	$0.483$	$1.166$	$0.952$	143/884	$0.934$	$0.612$	$0.23[0.20,0.27]$	$0.64[0.59,0.68]$
$11$	$199$	$00101010101$	$2$	$0.335$	$0.636$	$0.977$	56/819	$0.995$	$0.432$	$0.07[0.04,0.10]$	$0.42[0.38,0.47]$
$12$	$322$	$010101010101$	$1$	$0.439$	$1.201$	$0.958$	262/1789	$0.990$	$0.731$	$0.20[0.17,0.23]$	$0.77[0.71,0.80]$
$12$	$322$	$010101010101$	$2$	$0.340$	$0.632$	$0.980$	48/834	$0.998$	$0.436$	$0.06[0.04,0.09]$	$0.42[0.37,0.47]$
$13$	$521$	$0010101010101$	$1$	$0.477$	$1.170$	$0.953$	100/844	$0.926$	$0.653$	$0.19[0.15,0.23]$	$0.66[0.61,0.70]$
$13$	$521$	$0010101010101$	$2$	$0.336$	$0.635$	$0.978$	77/814	$0.997$	$0.530$	$0.12[0.08,0.15]$	$0.54[0.49,0.58]$
$14$	$843$	$01010101010101$	$1$	$0.439$	$1.201$	$0.958$	98/896	$0.980$	$0.729$	$0.16[0.12,0.20]$	$0.70[0.65,0.75]$
$14$	$843$	$01010101010101$	$2$	$0.340$	$0.632$	$0.980$	34/828	$0.996$	$0.409$	$0.04[0.02,0.07]$	$0.40[0.35,0.45]$
$15$	$1364$	$001010101010101$	$1$	$0.472$	$1.174$	$0.954$	156/1760	$0.904$	$0.671$	$0.13[0.11,0.15]$	$0.67[0.62,0.71]$
$15$	$1364$	$001010101010101$	$2$	$0.336$	$0.635$	$0.978$	46/800	$0.998$	$0.456$	$0.08[0.05,0.11]$	$0.49[0.44,0.54]$
$16$	$2207$	$0101010101010101$	$1$	$0.439$	$1.201$	$0.958$	157/1776	$0.978$	$0.689$	$0.14[0.11,0.17]$	$0.71[0.65,0.75]$
$16$	$2207$	$0101010101010101$	$2$	$0.340$	$0.632$	$0.980$	61/2576	$0.997$	$0.388$	$0.03[0.02,0.04]$	$0.42[0.37,0.48]$
$17$	$3571$	$00101010101010101$	$1$	$0.468$	$1.177$	$0.955$	114/1769	$0.905$	$0.611$	$0.10[0.08,0.12]$	$0.64[0.58,0.67]$
$17$	$3571$	$00101010101010101$	$2$	$0.337$	$0.635$	$0.978$	80/2644	$0.992$	$0.396$	$0.04[0.03,0.05]$	$0.40[0.35,0.46]$
$18$	$5778$	$010101010101010101$	$1$	$0.439$	$1.201$	$0.958$	108/1716	$0.968$	$0.675$	$0.10[0.08,0.13]$	$0.67[0.61,0.72]$
$18$	$5778$	$010101010101010101$	$2$	$0.340$	$0.632$	$0.980$	32/2558	$0.997$	$0.362$	$0.01[0.00,0.02]$	$0.39[0.34,0.45]$
$19$	$9349$	$0010101010101010101$	$1$	$0.465$	$1.179$	$0.955$	84/1708	$0.907$	$0.621$	$0.08[0.06,0.10]$	$0.67[0.61,0.70]$
$19$	$9349$	$0010101010101010101$	$2$	$0.337$	$0.634$	$0.978$	18/1671	$0.997$	$0.386$	$0.006[0.000,0.017]$	$0.40[0.35,0.46]$
$20$	$\mathrm{15\hspace{0.17em}127}$	$01010101010101010101$	$1$	$0.439$	$1.201$	$0.958$	66/1736	$0.964$	$0.658$	$0.06[0.04,0.09]$	$0.67[0.62,0.72]$
$20$	$\mathrm{15\hspace{0.17em}127}$	$01010101010101010101$	$2$	$0.340$	$0.632$	$0.980$	22/2576	$0.995$	$0.317$	$0.008[0.000,0.017]$	$0.34[0.30,0.40]$
$21$	$\mathrm{24\hspace{0.17em}476}$	$001010101010101010101$	$1$	$0.463$	$1.181$	$0.955$	103/1736	$0.901$	$0.617$	$0.11[0.09,0.14]$	$0.67[0.61,0.70]$
$21$	$\mathrm{24\hspace{0.17em}476}$	$001010101010101010101$	$2$	$0.337$	$0.634$	$0.979$	30/2534	$0.995$	$0.411$	$0.02[0.01,0.03]$	$0.41[0.36,0.47]$
$22$	$\mathrm{39\hspace{0.17em}603}$	$0101010101010101010101$	$1$	$0.439$	$1.201$	$0.958$	73/1718	$0.974$	$0.612$	$0.08[0.06,0.11]$	$0.64[0.57,0.67]$
$22$	$\mathrm{39\hspace{0.17em}603}$	$0101010101010101010101$	$2$	$0.340$	$0.632$	$0.980$	25/2537	$0.993$	$0.292$	$0.015[0.006,0.024]$	$0.30[0.26,0.38]$
$23$	$\mathrm{64\hspace{0.17em}079}$	$00101010101010101010101$	$1$	$0.461$	$1.183$	$0.956$	58/1630	$0.902$	$0.561$	$0.08[0.06,0.10]$	$0.60[0.54,0.63]$
$23$	$\mathrm{64\hspace{0.17em}079}$	$00101010101010101010101$	$2$	$0.337$	$0.634$	$0.979$	19/2520	$0.995$	$0.355$	$0.012[0.004,0.021]$	$0.38[0.33,0.44]$

Table 2. Table 3: Numerical and hardware results for bracelet states | [ z half ] ⟩ |[z^{\text{half}}]\rangle and | [ z MIS ] ⟩ |[z^{\text{MIS}}]\rangle product states, as detailed at the start of Appendix   C .

$N$	$\left|𝒱\right|$	State	Depth ($p$)	${\tau }_{\mathrm{eff}}$	$𝜸$	Naïve Count	Perfect	Emulation	EM (H)	EM (E)
$5$	$11$	$\left[00101\right]$	$38$	$15.708$	0.494, 0.074, -0.008, 0.019, -0.023, 0.211, -0.046 0.009, 0.388, 0.012, 0.056, -0.075, -0.040, 0.019 0.031, 0.015, -0.283, 0.709, 0.051, -0.011, -0.174 0.034, 0.034, 0.020, 0.035, -0.043, 0.033, -0.163 0.110, -0.170, -0.048, -0.122, 0.116, -0.029, 0.008 0.064, -0.225, 0.107	460/947	$1.000$	$0.994$	$0.57$ [0.52, 0.61]	$0.97$ [0.94, 0.99]
$6$	$18$	$\left[000101\right]$	$25$	$10.485$	0.109, 0.230, 0.002, 0.087, -0.285, 0.556, 0.141 0.108, 0.578, -0.244, 0.191, -0.060, -0.106, -0.304 -0.077, -0.108, -0.330, -0.162, -0.014, 0.526, 0.155 0.591, 0.284, -0.266, 0.104	453/940	$0.996$	$0.994$	$0.53$ [0.49, 0.57]	$0.98$ [0.95, 0.99]
$7$	$29$	$\left[0000101\right]$	$19$	$8.234$	-0.286, -0.024, 0.156, 0.335, 0.451, 1.000, 0.814 -0.008, -0.623, -0.354, 0.564, 0.230, 0.066, -0.640 -0.225, -0.349, -0.173, -0.221, -0.486	529/924	$0.960$	$0.943$	$0.66$ [0.62, 0.69]	$0.93$ [0.90, 0.95]
$8$	$47$	$\left[00010101\right]$	$47$	$19.320$	0.123, 0.300, -0.245, 1.259, 0.720, -0.236, -0.194 -0.003, -0.098, 0.006, 1.157, 0.730, -0.447, -0.100 -0.068, 0.092, -0.244, 0.405, 0.095, 0.168, 0.118 -0.369, -0.277, -0.255, -0.318, -0.342, -0.078, -0.090 -0.005, 0.066, -0.451, -0.102, 0.352, 0.252, -0.023 0.609, 0.361, -0.013, -0.287, -0.389, -0.114, -0.537 0.009, -0.276, -0.126, 0.509, 0.043	190/938	$0.984$	$0.970$	$0.19$ [0.15, 0.23]	$0.94$ [0.91, 0.96]
$9$	$76$	$\left[000010101\right]$	$41$	$16.708$	-0.026, -0.205, -0.368, -0.118, -0.337, -0.132, 0.319 -0.064, -0.066, 0.981, -0.275, -0.109, -0.011, -0.037 -0.208, -0.522, 0.924, 0.205, -0.071, 0.078, -0.050 0.201, 0.255, -0.030, -0.116, 0.037, 0.011, -0.025 0.016, 0.012, 0.270, 0.203, 0.059, 0.273, -0.044 1.012, 0.049, 0.288, -0.183, 0.799, 0.206	181/934	$0.970$	$0.797$	$0.22$ [0.19, 0.25]	$0.78$ [0.74, 0.81]
$10$	$123$	$\left[0000010101\right]$	$47$	$18.979$	1.204, -0.112, 0.011, -0.081, 0.427, 0.420, -0.922 0.020, 0.077, -0.402, 0.323, 0.834, -0.081, 0.417 -0.013, -0.368, 1.169, 0.916, -0.199, 0.341, 0.284 -0.046, -0.087, -0.264, 0.085, -0.548, 0.349, -0.217 0.102, 0.125, 0.135, 0.057, -0.022, 0.076, -0.371 -0.178, -0.153, 0.306, 0.758, -0.036, 0.006, 0.024 -0.035, 0.002, -0.039, 0.100, -0.020	98/907	$0.859$	$0.835$	$0.12$ [0.09, 0.15]	$0.81$ [0.76, 0.82]
$11$	$199$	$\left[00000010101\right]$	$48$	$19.670$	0.002, 0.274, 0.363, 0.147, 0.039, 0.054, -0.247 -0.058, -0.068, 0.066, 0.711, -0.184, 0.104, 0.929 0.098, 0.051, 0.191, -0.083, -0.227, 0.185, -0.208 -0.173, -0.173, -0.106, 0.004, -0.031, -0.232, -0.285 -0.218, -0.120, -0.197, -0.131, -0.287, -0.149, -0.044 -0.446, -0.271, -0.210, -0.290, -0.067, -0.157, -0.007 -0.069, 0.141, 0.056, -0.234, -0.177, 0.332	65/909	$0.751$	$0.643$	$0.07$ [0.05, 0.09]	$0.61$ [0.56, 0.64]
$12$	$322$	$\left[000001010101\right]$	$43$	$17.459$	0.983, 0.241, 0.820, 0.094, 0.044, 0.996, 0.136 -0.012, 0.072, 0.547, 0.165, 0.120, -0.004, -0.083 0.487, -0.094, 0.271, -0.016, 0.020, 0.281, 0.409 0.319, 0.681, -0.069, -0.061, 0.887, -0.056, -0.238 -0.505, -0.036, 0.073, -0.192, 0.535, -0.234, 0.073 -0.534, -0.750, -0.393, -0.169, -0.522, -0.436, 0.019 0.472	68/909	$0.580$	$0.568$	$0.10$ [0.07, 0.12]	$0.56$ [0.50, 0.58]
$5$	$11$	$\left[00101\right]$	$38$	$15.708$	0.494, 0.074, -0.008, 0.019, -0.023, 0.211, -0.046 0.009, 0.388, 0.012, 0.056, -0.075, -0.040, 0.019 0.031, 0.015, -0.283, 0.709, 0.051, -0.011, -0.174 0.034, 0.034, 0.020, 0.035, -0.043, 0.033, -0.163 0.110, -0.170, -0.048, -0.122, 0.116, -0.029, 0.008 0.064, -0.225, 0.107	460/947	$1.000$	$0.994$	$0.57$ [0.52, 0.61]	$0.97$ [0.94, 0.99]
$6$	$18$	$\left[010101\right]$	$24$	$10.095$	0.921, 0.081, -0.386, -0.371, -0.530, 0.084, 0.064 0.035, -0.321, 0.254, 0.245, 0.035, -0.056, 0.248 0.065, 0.009, -0.068, 0.102, 0.016, 0.103, 0.353 0.396, -0.422, -0.587	521/934	$0.996$	$0.986$	$0.72$ [0.67, 0.75]	$0.98$ [0.95, 0.98]
$7$	$29$	$\left[0010101\right]$	$24$	$9.925$	0.244, -0.004, -0.228, 0.175, 0.096, 0.023, -0.277 -0.143, -0.022, -0.300, 1.014, 0.060, -0.009, -0.176 -0.094, -0.212, 0.061, 0.056, 0.343, 0.190, 0.000 -0.164, -0.121, -0.242	456/936	$0.988$	$0.988$	$0.62$ [0.58, 0.66]	$0.96$ [0.93, 0.98]
$8$	$47$	$\left[01010101\right]$	$40$	$16.398$	0.826, -0.028, -0.126, -0.179, 0.098, -0.082, -0.030 -0.202, -0.426, -0.028, 0.686, 0.090, -0.212, 0.150 0.202, -0.386, -0.154, 0.122, 0.044, 0.225, -0.001 0.182, 0.127, -0.315, -0.191, 0.481, 0.259, -0.473 0.095, 0.977, 1.219, -0.070, 0.359, -0.027, 0.213 -0.088, -0.018, 0.212, 0.141, -0.035	259/920	$0.990$	$0.957$	$0.38$ [0.35, 0.42]	$0.95$ [0.91, 0.96]
$9$	$76$	$\left[001010101\right]$	$18$	$7.884$	1.435, 0.430, 0.101, 0.216, -0.042, 0.105, -0.295 0.352, -0.023, 0.099, -0.002, 0.212, 0.417, -0.052 -0.118, 0.243, -0.634, -1.151	542/895	$0.976$	$0.938$	$0.87$ [0.81, 0.88]	$0.92$ [0.88, 0.93]
$10$	$123$	$\left[0101010101\right]$	$34$	$14.027$	-0.869, -0.385, -0.024, 0.184, -0.497, 0.415, 0.155 0.084, 0.025, -0.128, 0.238, 0.162, 0.214, 0.964 0.872, 0.155, 0.056, 0.115, -0.170, -0.052, -0.037 0.192, 0.103, 0.148, 0.061, 0.061, 0.124, -0.244 -0.121, 0.063, -0.111, 0.035, 1.110, 1.184	272/911	$0.960$	$0.903$	$0.44$ [0.40, 0.48]	$0.88$ [0.84, 0.90]
$11$	$199$	$\left[00101010101\right]$	$10$	$4.572$	1.577, 0.240, -0.008, 0.129, -0.432, 0.005, -0.759 -0.388, -0.622, -1.846	513/913	$0.990$	$0.929$	$0.85$ [0.79, 0.86]	$0.93$ [0.88, 0.93]
$12$	$322$	$\left[010101010101\right]$	$26$	$10.825$	1.178, 0.160, -0.266, -0.031, 0.387, 0.053, 0.220 0.115, -0.098, -0.315, -0.460, -0.357, -0.380, -0.076 -0.167, -0.070, 0.228, 0.123, 0.036, 0.485, 0.331 0.258, 0.252, 0.086, -0.358, -1.016	213/883	$0.765$	$0.856$	$0.39$ [0.35, 0.43]	$0.83$ [0.79, 0.85]
$13$	$521$	$\left[0010101010101\right]$	$20$	$8.674$	-1.181, -0.232, -0.033, 0.031, 0.086, 0.149, -0.183 0.216, -0.231, 0.038, -0.242, 0.014, -0.107, 0.246 0.217, 0.356, 0.689, 0.745, 0.809, 1.815	369/893	$0.974$	$0.893$	$0.68$ [0.62, 0.70]	$0.89$ [0.83, 0.89]
$14$	$843$	$\left[01010101010101\right]$	$26$	$10.825$	-1.159, -0.479, -0.159, -0.025, -0.041, 0.027, 0.270 0.414, 0.449, 0.621, 0.533, 0.429, 0.264, -0.251 0.199, 0.416, 0.132, 1.116, 1.318, 0.926, 0.774 0.951, 0.156, 0.191, 1.008, 1.032	218/913	$0.845$	$0.875$	$0.41$ [0.36, 0.44]	$0.86$ [0.81, 0.87]
$15$	$1364$	$\left[001010101010101\right]$	$20$	$8.624$	1.492, 0.149, 0.268, 0.127, 0.035, -0.067, -0.073 0.060, -0.261, -0.003, 0.059, -0.452, 0.259, -0.545 -0.513, 0.329, 1.458, -0.224, 0.320, 0.192	183/876	$0.830$	$0.737$	$0.35$ [0.29, 0.37]	$0.74$ [0.67, 0.75]
$17$	$3571$	$\left[00101010101010101\right]$	$15$	$6.383$	0.972, 1.184, 0.297, 1.054, -0.300, 1.005, -0.772 -0.410, -0.740, -0.186, 0.058, 0.397, 0.054, 0.015 -0.583	182/876	$0.708$	$0.557$	$0.37$ [0.30, 0.38]	$0.52$ [0.45, 0.53]

Table 3. Table 4: Power-law fit parameters for amplification scaling A = c ​ | 𝒱 | α A=c\,|\mathcal{V}|^{\alpha} (see Eq.   13 ) across CTQW (Perfect), noiseless Rydberg emulation, and Aquila hardware. Values are listed for depths p = 1 , 2 , 3 p=1,2,3 for z half z^{\rm half} and p = 1 , 2 p=1,2 for z MIS z^{\rm MIS} ).

Target	Platform	$p$	$c$	$\alpha$	$R^2$
$z^{\mathrm{half}}$	CTQW	1	1.099	$0.878\pm 0.006$	0.9870
		2	1.112	$0.935\pm 0.003$	0.9997
		3	1.015	$0.981\pm 0.002$	0.9997
	Emulation	1	0.632	$0.877\pm 0.009$	0.9881
		2	0.719	$0.934\pm 0.006$	0.9961
		3	0.785	$0.930\pm 0.006$	0.9959
	Hardware	1	0.595	$0.729\pm 0.041$	0.9326
		2	0.463	$0.876\pm 0.025$	0.9374
		3	0.418	$0.734\pm 0.051$	0.7889
$z^{\mathrm{MIS}}$	CTQW	1	0.995	$0.996\pm 0.001$	0.9972
		2	1.000	$1.000\pm 0.000$	1.0000
	Emulation	1	0.641	$1.000\pm 0.004$	0.9838
		2	0.507	$0.967\pm 0.006$	0.9857
	Hardware	1	0.512	$0.820\pm 0.029$	0.9724
		2	0.285	$0.703\pm 0.067$	0.9250

Equations

$${\hat{U}}_W(\tau )=\exp (-\mathrm{i}\tau \hat{\mathcal{G}}),$$

$$\hat{\mathcal{G}}=\sum _{(a,b)\in \mathcal{E}}{w}_{ab}|a⟩⟨b|,$$

$$\hat{C}=\sum _{a\in \mathcal{V}}{c}_a|a⟩⟨a|.$$

$$|\psi ⟩=\prod _{q=1}^p{e}^{-i{\tau }_q\hat{\mathcal{G}}}{e}^{-i{\gamma }_q\hat{C}}|{\psi }_0⟩,$$

$$|{\psi }_0⟩\equiv |{\tau }_0⟩=e^{-\text{i}{\tau }_0\hat{\mathcal{G}}}|0⟩.$$

$$\hat{C}=\frac{1}{2}\sum _{i=1}^N{c}_i(\mathbb{I}-{\hat{\sigma }}_z^{(i)}).$$

$$d_h(z^a,z^b)\equiv \Vert z^a-z^b{\Vert }_1,$$

$$\mathcal{V}=\{z\in \{0,1{\}}^N|z_i+z_j\le 1\forall (i,j)\in E(G)\},$$

$$\hat{\mathcal{G}}=\sum _{i=1}^N\hat{\mathcal{P}}{\hat{\sigma }}_x^{(i)}\hat{\mathcal{P}},$$

$${\text{MAX}}_{\gamma ,\tau }:|⟨\gamma ,\tau |\psi ⟩{|}^2.$$

$$|[z]⟩=\frac{1}{\sqrt{|[z]|}}\sum _{u\in [z]}|u⟩,$$

$$\left|\right[010010001]⟩=\frac{1}{\sqrt{18}}(|010010001⟩+|010010100⟩+|010001010⟩+$$

$$|010001001⟩+|010100100⟩+|010100010⟩+|101001000⟩+$$

$$|101000100⟩+|100101000⟩+|100100010⟩+|100010100⟩+$$

$$|100010010⟩+|001010001⟩+|001010010⟩+|001001010⟩+$$

$$001000101⟩+|000101001⟩+|000100101⟩).$$

$$A(|\mathcal{V}|,p)=\frac{|\mathcal{V}|P(z^{\ast },p)}{|z^{\ast }|},$$

$$A(|\mathcal{V}|,p)=c|\mathcal{V}{|}^{\alpha },$$

$$n=\frac{1}{1-\alpha },$$

$$|\psi ⟩=\underset{k=0}{\overset{N-1}{⨂}}{e}^{-i{\tau }_1{\sigma }_x^k}{e}^{-i\gamma c_k{\sigma }_z^k}{e}^{-i{\tau }_0{\sigma }_x^k}|0⟩.$$

$${\hat{U}}_{z^{\ast }}(\gamma )|z⟩=\exp \left(-i\gamma \sum _{i:z_i^{\ast }=1}{z}_i\right)|z⟩$$

$$\mathcal{G}={\mathcal{G}}_{+}+{\mathcal{G}}_{-},$$

$${\mathcal{G}}_{\pm }=\frac{1}{2}\left(\mathcal{G}\pm U_{z^{\ast }}\mathcal{G}{U}_{z^{\ast }}\right),$$

$$R_{\pm }(\tau )=e^{-i\tau ({\mathcal{G}}_{+}\pm {\mathcal{G}}_{-})},$$

$$|{\tau }_0,{\tau }_1⟩=[R_{-}({\tau }_1)R_{+}({\tau }_1){]}^{p^{\prime }}|{\psi }_0⟩,$$

$$R_{-}({\tau }_1)R_{+}({\tau }_1)=\exp [-2i{\tau }_1{\mathcal{G}}_{+}-{\tau }_1^2[{\mathcal{G}}_{+},{\mathcal{G}}_{-}]+O({\tau }_1^3)].$$

$$|v_j⟩=\frac{1}{\sqrt{|S_j|}}\sum _{z\in S_j}|z⟩,$$

$$J_{j,j+1}=J_{j+1,j}=\sqrt{(k-j)(j+1)},$$

$$J_{0\to z^{\ast }}={\left(\prod _{j=0}^{k-1}{J}_{j,j+1}\right)}^{1/k}.$$

$$⟨v_k|e^{-i{T}_{\mathrm{eff}}J}|v_0⟩\approx \{\begin{array}{ll}-i\sin \left(T_{\mathrm{eff}}{J}_{0\to z^{\ast }}\right),& k\text{odd},\\ 1-\cos \left(T_{\mathrm{eff}}{J}_{0\to z^{\ast }}\right),& k\text{even},\end{array}$$

$$\mathrm{Prob}(|z^{\ast }⟩)\propto {\sin }^2\left(T_{\mathrm{eff}}{J}_{0\to z^{\ast }}\right),$$

$$|{\psi }_0⟩=|v_0⟩-i{\tau }_0\left({\beta }_{+}|v_1⟩+{\beta }_{-}|w_1⟩\right)+\mathcal{O}({\tau }_0^2).$$

Full text

I Introduction

Quantum search and optimization are key applications for near-term quantum computers [50]. Algorithms for unstructured or spatial search and the Quantum Approximate Optimization Algorithm (QAOA) exemplify the unique properties of quantum computing: superposition to explore exponentially many computational pathways in quantum parallel, entanglement to encode a cost landscape, and interference to amplify desired states [30, 3]. These algorithms typically utilize an ansatz111A parameterised family of circuits used to prepare candidate states, with the parameters chosen (typically by a classical outer loop) to maximize the probability of sampling desirable solutions. comprised of an interleaved sequence of unitaries. At each ansatz iteration, the first diagonal unitary phase-encodes the cost or validity of each solution state, and the second mixing unitary drives amplitude transfer between these states. This layered structure can be naturally understood through the framework of continuous-time quantum walks (CTQWs), where coherent dynamics of quantum states traversing a graph underpin the convergence behavior of the algorithm.

CTQWs provide a natural framework for quantum algorithm design, with coherent dynamics over a graph underlying methods for spatial search and optimization [19, 18]. In particular, CTQW-based search algorithms achieve the optimal quadratic improvement in query complexity for unstructured search [3], and operate in a non-adiabatic regime that can saturate fundamental quantum speed limits [23, 22].

These algorithms can all be expressed as a phase-walk ansatz, an alternating sequence of two interleaved unitaries. With each ansatz iteration a phase separator applies rotations that encode the problem at hand, and a mixing unitary carries out a CTQW that drives amplitude transfer between states. While the query-complexity bound for unstructured search is quadratic, the phase-encoding layer can carry additional information about global problem structure. When symmetries of the mixing graph (whose edges define allowed transitions) reflect the structure of the solution space, compounding interference across symmetric pathways can coherently focus amplitude on target states, yielding super-quadratic convergence [42, 8]. Mixing graphs that enable such “shortcut” dynamics are especially significant for near-term quantum devices, where practical advantage requires sufficient amplification within time-limited coherence windows.

Continuous-time quantum walks have been demonstrated in the position basis across a variety of platforms, including walks over line graphs using photonic waveguides [47], neutral atoms [49] and trapped ions [58], as well as two-dimensional grids using superconducting qubit arrays [65, 32] and tweezer-programmable strontium-atom lattices [66]. Beyond spatial walks, variational circuits that implement CTQWs in the computational basis have been demonstrated with photonic waveguides [51], and CTQW-like dynamics have been observed in quantum many-body scar dynamics in Rydberg chains [21] on neutral atom systems [9]. Full phase-walk ansätze with alternating phase-encoding and mixing unitaries have been implemented for QAOA on superconducting processors [37] and trapped ions [57].

Despite this progress, each approach addresses only part of the phase-walk framework. Position-basis implementations demonstrate coherent quantum dynamics but cannot leverage the full computational-basis Hilbert space and, as the walk graph is fixed by hardware geometry, are not easily tailored to match the structure of a target problem. Gate-based QAOA implementations on superconducting and trapped-ion processors do include phase encoding. However, the mixing unitary is typically limited to a transverse-field operator on an unconstrained Hilbert space, requiring either penalty terms or post-selection to enforce feasibility constraints– both of which incur overhead that grows with system size [36]. Recent QAOA studies have reported evidence of scaling advantages over classical algorithms, but these claims derive entirely from noiseless classical simulations, with hardware demonstrations showing only Grover-like amplification at low circuit depth [15, 57].

Analog neutral-atom platforms provide an ideal environment for the exploration of phase-walk ansätze. The blockade interaction, whereby strong Rydberg-state repulsion suppresses simultaneous excitation of nearby atoms, enables efficient implementation of constrained CTQWs whose walk graphs possess non-trivial topology with direct relevance to constrained combinatorial optimization [36, 41, 31]. The analog evolution has the potential to natively implement continuous-time walk dynamics, while global and local detuning provide flexible means to apply site-dependent phase shifts encoding the cost function. This combination yields a hardware-efficient realization of the full phase-walk ansatz in which both constraint enforcement and the mixing unitary are “free” in terms of gate overhead.

In this work, we report the implementation of continuous-time quantum walk (CTQW)-based variational ansätze on analog-mode neutral-atom hardware, using QuEra’s Aquila processor [61]. Unlike prior Rydberg-atom demonstrations that sample low-energy configurations or observe emergent many-body dynamics [26, 9], we target the preparation of specific, known benchmark states and systematically quantify convergence as a function of ansatz depth. This provides a controlled setting that isolates the intrinsic performance of CTQW-based dynamics from problem-dependent complexity.

We focus on two classes of target states: computational-basis product states, and “bracelet” states–equal-weight superpositions over all basis states related by the dihedral symmetries of the walk graph. The former relies on dynamics that are analogous to the preparation of an unknown unentangled state in search or optimization. The latter generalize Dicke states, the symmetric entangled states widely used in quantum metrology and constrained optimization [6, 36], to an independent-set subspace. While Dicke states span a subspace that grows only linearly with system size, bracelet states live in an exponentially large symmetric sector, making their preparation a more demanding test of coherent control. Furthermore, their preparation probes whether the non-adiabatic dynamics for preparation of product states in constrained Hilbert spaces [39] via quantum many-body scar dynamics [9, 21] extend to the preparation of similarly constrained entangled targets.

For product-state preparation, we derive closed-form expressions for near-optimal walk parameters, providing analytically guided initial points via a method that is generalizable to walk graphs defined by independent-set subspaces. For bracelet states, we introduce an optimization protocol based on the spectral properties of the walk dynamics [38]. By comparing ideal CTQW predictions, noiseless Rydberg emulation, and measurement from noisy hardware, we identify the depth at which super-quadratic amplification persists and the crossover beyond which noise suppresses convergence, providing a practical figure of merit for NISQ implementations of CTQW-based algorithms. Finally, we employ quench dynamics under the walk Hamiltonian to probe coherence of the prepared bracelet states, providing empirical evidence that the observed populations arise from coherent superposition rather than an incoherent mixture.

The remainder of this work is as follows. In Section II, we introduce the phase-walk ansätze, performance metrics used in later analysis, the representative target states, and the variational optimization strategies. In Section III, we describe the mapping of the ansätze to the Rydberg Hamiltonian on Aquila. Results of state preparation on hardware are compared with the scaling predicted by ideal CTQW dynamics and noiseless Rydberg emulation in Section IV. In Section V, we apply quenches on the prepared states as an indicator of coherent state preparation. Finally, in Section VI, we summarize our conclusions. Overall, our results establish constrained-subspace CTQWs as a framework for entangled state preparation on analog-mode neutral-atom processors, and demonstrate that CTQW-based ansätze for known states can exhibit the super-quadratic convergence characteristic of efficient quantum-walk protocols on currently available hardware.

II Phase-walk ansätze

II.1 Quantum Walk-based Ansätze

A Continuous-Time Quantum Walk (CTQW) [60] is the coherent evolution of a quantum state under the unitary time-evolution operator

[TABLE]

where the generator

[TABLE]

is obtained from the (possibly weighted) adjacency matrix of a quantum walk graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with vertex set $\mathcal{V}$, edge set $\mathcal{E}$, and real, symmetric weights $w_{ab}=w_{ba}$. Each vertex is matched to a basis state of a quantum state $a\leftrightarrow|a\rangle$, and the terms of the graph adjacency matrix are matched to the generator.

In the context of algorithms for search and optimization, CTQWs are often paired with a diagonal unitary that adds phase to the graph vertices weighted by $\phi_{a}=\gamma c_{a}$, defined by a (variational) scaling parameter $\gamma$ and the generator

[TABLE]

Together the two unitaries form a parameterized phase-walk ansatz for which a common task is to prepare some target or resource state by alternating between the walk (or “mixer”) generator $\hat{\mathcal{G}}$ and phase-separator generator $\hat{C}$ applied to some fiducial initial state $|\psi_{0}\rangle$

[TABLE]

for $2p$ variational parameters $\{\vec{\gamma},\vec{\tau}\}$ – as exemplified by the quantum approximate optimization algorithm [30].

A common representation of basis states is binary bitstrings $a\in\mathcal{Z}=\{0,1\}^{n}$, where each bit $z_{i}$ represents the state of qubit $i$. In this work, we define $|\psi_{0}\rangle$ as an initial walk from the all-zeros state,

[TABLE]

The phase-separator generator can be written as some function $c_{a}\equiv f(z)$, typically a low-order polynomial to conform to the locality constraints of quantum hardware. Here, we choose a first-order function $f(z)=\sum_{i}c_{i}z_{i}$ so that the phase-separator generator can be efficiently written in terms of Pauli matrices

[TABLE]

Similarly, the walk graph is often formed by edges between vertices whose labels differ by a Hamming distance of one or two [30, 36, 42, 8]:

[TABLE]

Hamming-distance-based constructions yield graphs that are efficiently implementable and typically highly symmetric (generally vertex-transitive) [42, 8]. If edges connect labels with $d_{h}=1$ over all $2^{N}$ possible bitstrings, then the walk graph is the $N$-dimensional hypercube. A Hamming-distance-1 graph walk generator can be efficiently written as $\hat{\mathcal{G}}=\sum_{i}\hat{\sigma}_{x}^{(i)}$, and small Hamming distances correspond to low-depth polynomial functions over Pauli operators.

II.2 Walks in constrained subspaces

Frameworks for optimization with constraints restrict the vertices to a subset of valid solutions and construct the walk generator so that it preserves this reduced search space as an invariant subspace [41, 8, 36]. The resulting CTQW is often non-separable and inherits the symmetries of the constraint, while breaking others. As we detail in Section III, such walks are the focus of this work as neutral-atom platforms offer unique opportunities for their efficient implementation. We constrain the full space of length-$N$ bitstrings according to an underlying constraint graph $G=(V,E)$ on $N$ vertices. The walk vertices $\mathcal{V}$ are then independent-set configurations of $G$,

[TABLE]

with edges between vertices in $\mathcal{V}$ having $d_{h}=1$. The resulting walk graph is a subgraph of the hypercube, and the walk generator can be written efficiently as

[TABLE]

where $\hat{\sigma}_{x}^{(i)}$ is the Pauli-$X$ operator on qubit $i$, and $\hat{\mathcal{P}}$ is the projector onto $\mathcal{V}$ (equivalently, the independent-set subspace of the constraint graph $G$).

In particular, we choose the constraint graph $G$ to be the cycle graph on $N$ vertices (i.e., a ring with nearest-neighbor edges). The resulting walk graph is a Lucas cube [45], shown in Fig. 1. We make this choice because its structural properties are relatively well understood, supporting efficient enumeration and generation of $\mathcal{V}$, and reduced-dimensional representations of $\mathcal{G}$, at large $N$ (which is not the case for all independent-set subspaces [59]), while remaining suitably complex to produce non-trivial dynamics. Specifically, the number of vertices of the Lucas cube is exponentially smaller than that of the hypercube at the same $N$, yet still grows exponentially as $|\Lambda_{N}|\sim\varphi^{N}$, where $\varphi=(1+\sqrt{5})/2$ and, as it preserves the dihedral symmetry of $G$, it retains a relatively small but structurally rich dihedrally symmetric sector that grows like $B_{N}^{(D)}\sim\varphi^{N}/(2N)$ [4].

II.3 State preparation

A key application of quantum walks is to prepare defined target states $|\psi\rangle$. Commonly, this is some unknown but well-defined state, such as a product state that maximizes the value of some low-order polynomial objective $C(z)$, such as in QAOA [30]. Similarly, the states could be defined concretely as a resource state with an efficient description, such as the GHZ state $|\psi\rangle=(|00\cdots 00\rangle+|11\cdots 11\rangle)/\sqrt{2}$. Given the extensive literature on QAOA and its variants to solve combinatorial objectives $C(z)$, this work focuses instead on preparing concretely defined states and characterizing the capacity of variational quantum walks to prepare these target states on blockaded subspaces [33].

Given some defined target state $|\psi\rangle$, a variational optimizer chooses $p(N+1)$ parameters $\{\vec{\gamma},\tau\}$ of the variational ansatz state $|\gamma,\tau\rangle$ to maximize the overlap with the target state

[TABLE]

Or, similarly, minimize the statistical divergence between the target distribution $P(z)=|\langle z|\psi\rangle|^{2}$ and the variational distribution $Q(z\,|\,\gamma,\tau)=|\langle z|\gamma,\tau\rangle|^{2}$.

II.4 Target states

For the constraint graph of the $N$-vertex ring, we chose two classes of target states. The first class are product states, e.g. $|\psi\rangle=|0010101\rangle$ specified by a bitstring $z$. While preparing such states for a system of non-interacting qubits is trivial (see Section II.6), preparing these known states using walk-based dynamics provides a benchmark for the typical goal of variational ansätze: preparing unknown, low-entanglement states corresponding to low-cost solutions [30, 10, 36]. Furthermore, because the mixer allows only constraint-respecting transitions, it is not a priori clear that arbitrary product states can be efficiently prepared with high fidelity using an ansatz parameterised solely by walk times and locally tuned phase-separator values $\gamma_{i}$. We focus on two representative product-state targets. The first, denoted $z^{\rm half}$, is the state $|0^{N-2h}(01)^{h}\rangle$ with $h=\lfloor N/2\rfloor$, corresponding to roughly half the maximum Hamming weight for the ring graph. The second, denoted $z^{\rm MIS}$, is a maximum independent set of the ring: the alternating pattern $(01)^{N/2}$ for even $N$, or $0(10)^{(N-1)/2}$ for odd $N$.

The second class consists of superpositions over the symmetric subspace. An $N$-vertex ring has translational and reflective symmetries described by the dihedral group $D_{N}$ (i.e., the set of all discrete rotations and reflections). Because the constraint graph $G$ is defined on the ring, these operations relabel vertices without changing which pairs are adjacent (which are referred to in graph theory as automorphisms of $G$). We therefore consider the corresponding symmetry-invariant subspace, spanned by equal-weight superpositions of all computational-basis states labeled by bitstrings related by cyclic rotations or reflections,

[TABLE]

where $[z]=\{g\!\cdot\!z:\,g\in D_{N}\}$ is the dihedral orbit of a representative bitstring $z$ 222The first instance in which the symmetric subspace contains states whose entire orbit cannot all be generated by rotations alone occurs at $N=9$, where

$\displaystyle\big|\big[010010001\big]\big\rangle=\,\tfrac{1}{\sqrt{18}}\big(|010010001\rangle+|010010100\rangle+|010001010\rangle+$

$\displaystyle|010001001\rangle+|010100100\rangle+|010100010\rangle+|101001000\rangle+$

$\displaystyle|101000100\rangle+|100101000\rangle+|100100010\rangle+|100010100\rangle+$

$\displaystyle|100010010\rangle+|001010001\rangle+|001010010\rangle+|001001010\rangle+$

$\displaystyle 001000101\rangle+|000101001\rangle+|000100101\rangle\big).$

.

We refer to these states as bracelet states, borrowing from the combinatorial term for equivalence classes of bitstrings under cyclic rotations and reflections [55]. The preparation of such states is an informative benchmark, as it requires coherent many-body interference to establish uniform amplitudes across a dihedral orbit of up to $2N$ basis states. Furthermore, bracelet states have potential applications as initial states for constrained optimization [36, 7] and quantum metrology [6].

II.5 Ansätze performance

Phase-walk ansätze are heuristic methods for preparing approximate solutions to combinatorial problems. Outside a few special cases, formal guarantees are limited, so benchmarking is primarily empirical and problem-dependent [30, 41, 63]. In this context, an informative metric is the amplification of desirable solutions relative to a uniform baseline over the accessible subspace

[TABLE]

where $P(z^{*},p)$ is the probability of preparing the target $|z^{*}\rangle$ at depth $p$ [42, 8] and $|z^{*}|$ is the cardinality of the target set of basis states. Effective ansätze are expected to closely follow a power-law growth in $|\mathcal{V}|$,

[TABLE]

with the exponent $\alpha$ as the key performance indicator [16, 2]. Because the single-shot success scales as $|\mathcal{V}|^{\alpha-1}$, $\alpha$ induces an effective polynomial speedup order $n$,

[TABLE]

which can be understood as a proxy for query-complexity scaling (e.g., at $\alpha=\tfrac{1}{2}$ the scaling is Grover-like).

For a particular phase-walk ansatze to offer the possibility of quantum-speedup given the overheads of optimization, sampling, and classical post-processing, it should ideally demonstrate two key features. First, at fixed depth $p$, the amplification factor $A$ should grow with problem size as a high-order polynomial in $|\mathcal{V}|$. Second, at fixed $|\mathcal{V}|$, the amplification should itself increase polynomially with circuit depth $p$. In both instances, orders $n>2$ indicate the leveraging of global phase-structure to accelerate convergence [42, 67].

In the following, we probe these criteria through the preparation of known target states, seeking a reliable protocol for CTQW implementation and an empirical view of potential quantum speedup possible within the degrees of freedom of our ansätze when applied to non-trivial problem instances.

II.6 Variational optimization

II.6.1 Product states

If the constraint graph $G$ is the null graph, then the walk Hamiltonian is the $N$-dimensional hypercube and the time evolution remains fully separable. With $p=1$, the state is

[TABLE]

Setting the $X$-rotation angles to $(\tau_{0},\tau_{1})\equiv\{\pi/4,\,3\pi/4\}\pmod{2\pi}$ (in any order) and choosing $\gamma=\pm\pi\pmod{2\pi}$ and $c_{k}=z_{k}^{*}$ enables the preparation of any computational-basis state333Geometrically, the single-qubit gates perform an $X\text{-}Z\text{-}X$ rotation sequence on the Bloch sphere of each qubit. Starting from $+z$ (i.e., $|0\rangle$), an $X$-rotation by $\tau_{0}=\pi/4$ moves the Bloch vector to $-y$. Then, if $c_{k}=1$, the intermediate $Z$-rotation moves the vector to $-y$. Finally, an $X$-rotation by $\tau_{1}=3\pi/4$ drives vectors on $+y$ back to $+z$ (i.e., $|0\rangle$) and those on $-y$ to $-z$ (i.e., $|1\rangle$).. However, when a nontrivial connected constraint graph is introduced, each $\sigma_{x}^{k}$ becomes a neighbor-conditioned multi-qubit gate, so the evolution is no longer separable. Nevertheless, the walk ansatz can still prepare the exact state. Given a phase separator

[TABLE]

as a sum of Pauli-z terms that are non-zero in the target bitstring, three protocols are possible. The first is a Trotterized adiabatic protocol that uses perturbatively small parameters $\gamma,\tau\ll 1$, exact in the $p\to\infty$ limit [62]. The second is a Trotterized mask protocol with $\gamma,\tau\ll 1$ but $\gamma\gg\tau$ to “pin” the zero qubits in the zero state with a large effective Z field. The third is a fast-forward limit [35] with $\gamma=\pi$ and $\tau_{0}+p\tau_{1}=\tfrac{\pi}{2}$, for which we give the following argument. First, the walk Hamiltonian can be split by $U_{z^{*}}$ into two components:

[TABLE]

where $\mathcal{G}_{-}$ flips $h_{z^{*}}(z)$ and $\mathcal{G}_{+}$ preserves it. Then,

[TABLE]

Here the $+1$ eigenspace of $U_{z^{*}}$, which contains the ground state $|0\rangle$ and $|z^{*}\rangle$, is the positive subspace and the $-1$ eigenspace the negative subspace.

The interleaved $U_{z^{*}}$ effectively toggles a rotation between the negative and positive subspaces,

[TABLE]

which together produce constructive interference on the positive subspace. To illustrate this effect, consider the even-depth ($2p^{\prime}=p$) ansatz,

[TABLE]

where $|\psi_{0}\rangle$ is defined as in Eq. 5. By BCH expansion to second-order, a single pair is,

[TABLE]

The linear $\mathcal{G}_{-}$ terms cancel, so to leading order the dynamics are generated by $\mathcal{G}_{+}$ with the effective time $p^{\prime}\tau_{1}$.

Projection of $\exp(-i2\tau_{1}\mathcal{G}_{+})$ onto the positive subspace yields an effective evolution over states

[TABLE]

where $S_{j}$ is the set of bitstrings in the positive subspace with Hamming weight $j$, and the $|v_{j}\rangle$ are the corresponding uniform “shell” states of fixed Hamming weight (so $|v_{0}\rangle=|0\rangle$ and $|v_{k}\rangle=|z^{*}\rangle$). The effective Hamiltonian $J$ is the SU(2) spin-$k/2$ chain Hamiltonian (permutation-symmetric noninteracting spins $1/2$) expressed in the symmetric Dicke basis, so that in this subspace it reduces to a one-dimensional ladder $\{|v_{j}\rangle\}_{j=0}^{k}$, which has non-zero entries only on the first off-diagonals:

[TABLE]

since each state in $S_{j}$ has $k-j$ forward neighbors in $S_{j+1}$ and each state in $S_{j+1}$ has $j+1$ backward neighbors in $S_{j}$. Because $J$ only couples neighboring Hamming-weight shells $j\leftrightarrow j+1$, the first non-vanishing contribution connecting $|v_{0}\rangle$ to $|v_{k}\rangle$ appears at order $k$. The lowest-order coupling between $|0\rangle$ and $|z^{*}\rangle$ then occurs at order $k$ with strength,

[TABLE]

Moreover, since $J$ inherits the bipartite structure of $\mathcal{G}$, the transition amplitude $\langle v_{k}|e^{-iT_{\rm eff}J}|v_{0}\rangle$ has a fixed parity in $T_{\rm eff}$ determined by $k$, and,

[TABLE]

which, in either case, gives

[TABLE]

maximized at $T_{\rm eff}=\tfrac{\pi}{2J_{0\rightarrow z^{*}}}$, with leakage proportional to $||\tau_{1}^{2}[\mathcal{G}_{+},\mathcal{G}_{-}]||$.

As $T_{\rm eff}$ is inversely proportional to $J_{0\rightarrow z^{*}}$, at low $k$ and $p^{\prime}$ we seek an initial state that reduces the second-order leakage. Let $|w_{1}\rangle=\tfrac{1}{\beta_{-}}\mathcal{G}_{-}|v_{0}\rangle$, with $\beta_{-}=||\mathcal{G}_{-}|v_{0}\rangle||$, and $\beta_{+}=||\mathcal{G}_{+}|v_{0}\rangle||$. The state prepared by an initial walk over $\mathcal{G}$ is

[TABLE]

To first order, the amplitude driving the chain is reduced by

[TABLE]

Consequently, the coupling $J_{0\rightarrow z^{*}}$ is attenuated to the effective coupling $J_{\rm eff}(\tau_{0})=J_{0\rightarrow z^{*}}\cos(\phi)$, with

[TABLE]

Minimization of the second-order leakage leads to a unique positive-valued solution for $\tau_{0}$ and $\tau_{1}$. The second-order term $-\tau_{1}^{2}[\mathcal{G}_{+},\mathcal{G}_{-}]$ returns amplitude from $|w_{1}\rangle$ to the positive subspace proportional to $\tau_{1}^{2}\,\kappa_{\rm ret}\,\sin\phi$ and leaks amplitude from $|v_{1}\rangle$ proportional to $\tau_{1}^{2}\kappa_{\rm leak}\cos\phi$, where $\kappa_{\rm ret}$ and $\kappa_{\rm leak}$ are graph-dependent constants. These channels are balanced when

[TABLE]

As

[TABLE]

to first-order the optimal leakage-reducing $\tau_{0}$ is

[TABLE]

where $\kappa=\kappa_{\rm leak}/\kappa_{\rm ret}$, and

[TABLE]

For odd $p$, Eq. 32 still holds, but the absence of a full $+/-$ pair leaves a linear $\mathcal{G}_{-}$ contribution. The resulting leakage scales as $\mathcal{O}\big(\tau_{1}-\tau_{0}\big)$, rather than $\mathcal{O}(\tau_{1}^{2})$ as in the even (paired) case. In practice, this can be mitigated by further attenuating $J_{\rm eff}$ via a larger $\tau_{0}$. More generally, while higher-order terms contribute significantly if $\tau_{0},\tau_{1}\not\ll 1$, near $(\tau_{0}^{*},\tau_{1}^{*})$, quadratic curvature from second-order terms still dominates both the linear odd-$p$ residual leakage and any higher-order corrections. The result is a locally smooth and convex objective function – with local optimization initiated at $(\tau^{*}_{0},\tau^{*}_{1})$ reliably converging to the unique optimum444For the unconstrained hypercube, $[\mathcal{G}_{+},\mathcal{G}_{-}]=0$. Consequently $R_{-}(\tau_{1})R_{+}(\tau_{1})=\exp(-2i\tau_{1}\,\mathcal{G}_{+})$ exactly, and perfect transfer is trivial at $p^{\prime}=1$ with $\tau_{0}=0$..

Figure 2 shows the result of running Nelder-Mead from the analytic starting point $(\tau_{0}^{*},\tau_{1}^{*})$ for system sizes $N=5$ to $23$ at depths $p=1$ to $5$. In every case the solver converges to an optimum in a single basin. For the largest system $N=23$, the worst-case success probability over all bit-string targets at $p=1$ is only 0.267 (occurring at Hamming weight $h=8$), but this minimum probability rises to 0.925 by $p=5$. Conversely, the $h=11$ state achieves a probability of 0.901 at $p=1$ and essentially unity at $p=5$. At fixed $N$ the probability profile as a function of Hamming weight is well described by a skewed quadratic, with its lowest point near $h\approx\lfloor N/4\rfloor$ and monotonic increase towards both the all-zero and maximum Hamming weight. The $(\tau_{0},\tau_{1})$ for the optimum depicted in the right panel are listed in Table 2. Each of the three depicted instances show $(\tau_{0}^{*},\tau_{1}^{*})$ within the locally convex region of the optima. Over $p=1$ to $3$ the effective walk time $T_{\rm{eff}}$, as given by Eq. 28, approaches $\pi/2$.

II.6.2 Bracelet states

The optimization strategy for preparation of bracelet states is guided by two practical considerations, as expanded in Section III. First, we aim to minimize the total evolution time and, second, utilize the hardware-native realization of the phase separator

[TABLE]

where $\hat{n}=\sum_{i=1}^{N}|1\rangle_{i}\langle 1|_{i}$ applies a uniform phase to all computational basis states with the same Hamming weight.

The initial stage sweeps the walk generator alone,

[TABLE]

at discrete times $\tau_{j}=j\Delta\tau$ for $0<\tau_{j}\leq\tau_{\rm{max}}$. At each sample we measure the population of the target bracelet subspace $\left|\langle[z^{*}]|\psi(\tau_{j})\rangle\right|^{2}$ and locate the local maxima $\tau^{(m)}_{\rm{peak}}$ higher than $1/|D_{N}|$, as demonstrated in the top panel of Fig. 3.

Starting from the second identified peak, we fix the total walk duration to $\tau_{\rm{tot}}=\tau^{(m)}_{\rm{peak}}$ and set the ansatz depth and walk times to,

[TABLE]

where $\tau_{\rm{min}}^{\rm{hw}}$ is the minimum allowed walk time ($\sim 0.4$) and the $-2$ guards against quantization error arising through hardware constraints so the walk times are fixed and only the phase separator $\vec{\gamma}=\{\gamma_{k}\}_{k=1}^{p}$ remain variational. As the resulting landscape is non-convex, we optimize these with the derivative-free COBYLA optimizer initialized at $\gamma_{k}=0$, stopping the scan over peaks when there is a decrease in success probability or the range of identified peaks is exhausted.

We discard the first peak because the state is concentrated in the $C_{N}$-symmetric (translation-invariant) subspace. Since the phase separator depends on Hamming weight, it acts as $e^{i\gamma h}\mathbb{I}$ on each weight-$h$ sector and therefore cannot distinguish distinct $D_{N}$-symmetric components. The total walk time $\tau_{\rm{tot}}$ must be long enough to accumulate relative phase between $D_{N}$-symmetric components at fixed $h$ before optimization over $\vec{\gamma}$ can appreciably improve the success probability.

More generally, the required $\tau_{\rm eff}$ for high-fidelity preparation of $|[z^{*}]\rangle$ is explainable with a frequency-resolution model. Let $\{(\lambda_{r},|r\rangle)\}_{r=1}^{d}$ be the eigenpairs of the walk generator $\mathcal{G}$ restricted to the $D_{N}$-invariant subspace [38]. Because the only interleaved control is the phase separator, resolvability of the target state is governed by spectral gaps $\lambda_{rs}=|\lambda_{r}-\lambda_{s}|$ between eigenmodes $r,s\in\{1,\dots,d\}$ that both have appreciable support on the same Hamming-weight sector as the target [5]. Consequently, control under variation of $\vec{\gamma}$ requires order-unity accumulated phase, $\lambda_{rs}\,\tau_{\mathrm{eff}}=\Theta(1)$.

We encode this threshold by introducing the dimensionless constant $\kappa=\Theta(1)$ and enforcing $\lambda_{rs}\tau_{\rm eff}\geq\kappa$. Among resolvable pairs, we then define the resolvable spectral minimum,

[TABLE]

and the model predicts that this low-frequency component sets the timescale,

[TABLE]

To identify which gaps are associated with the target Hamming-weight, $h^{*}=h([z^{*}])$, we expand each eigenmode $|r\rangle$ in the bracelet basis $\{|[z]\rangle\}$ and compute its weight in the target sector

[TABLE]

with $\Delta_{\min}(\kappa)$ chosen from the eigenmodes with non-zero weight.

The bottom panel of Fig. 3 shows the accumulated walk time $\tau_{\rm{eff}}$ required to reach high-fidelity preparation against $1/\Delta_{\rm min}(\kappa^{*})$ for all dihedrally distinct bracelet targets $|[z^{*}]\rangle$ at $N=5$ to $12$. These numerical results are consistent with the scaling relationship predicted by the frequency-resolution model. Across all targets, $\tau_{\rm{eff}}$ scales linearly with $1/\Delta_{\rm min}(\kappa^{*})$ for a single fixed $\kappa^{*}$, consistent with Eq. 37. This $1/\Delta_{\rm min}$ scaling stands in contrast to adiabatic state preparation, where the required evolution time scales as $1/\Delta_{\rm min}^{2}$ [1]. The improvement arises because the phase-walk ansatz operates in a non-adiabatic regime, creating a superposition of eigenmodes whose relative phases evolve and interfere constructively on the target rather than adiabatically tracking a single instantaneous ground eigenstate.

Within each $N$, targets that are a maximal independent set consistently exhibit the smallest $\tau_{\rm eff}$ and the largest $\Delta_{\min}$. This is consistent with prior work on scarred quantum walks [25], which places these states on or near a small, low-degree hypercube-like subgraph weakly coupled to the rest of the graph – with dynamics confined to this subgraph producing sparser, quasi-regular spectra with larger internal gaps, resulting in faster preparation.

III Rydberg atom arrays

Neutral atom quantum computers are a promising modality for various applications, from gate-based computing [54, 44], fault-tolerant algorithms [13, 52, 53, 14], and Hamiltonian simulation [56, 29, 34]. In a neutral atom computer, individual atoms, such as Rubidium 87, are trapped in optical lattices with the qubit encoded into the electronic states of valence electrons [28]. Quantum operations are implemented by manipulating the electronic state of the atom using finely tuned lasers. Entanglement is mediated by the Rydberg state, a highly excited orbital that strongly interacts via a van der Waals interaction with nearby atoms in the Rydberg state.

In this work, we focus on analog mode neutral atom computers, and specifically that of Aquila, QuEra Computing’s cloud analog Hamiltonian simulator [61]. Here, the qubit is encoded into a ground state $|g\rangle=|5S\frac{1}{2}\rangle$ and Rydberg state $|r\rangle=|70S\frac{1}{2}\rangle$ of Rb87. Due to adjacent Rydberg states constantly interacting, an analog mode computer implements the time-dependent dynamics of an Ising-like Hamiltonian

[TABLE]

The first term is a Rabi drive, which coherently drives each atom between the ground and Rydberg state. The second term is the detuning, which is a Pauli-Z-like term that applies an energy penalty to the Rydberg state. The detuning is separated into a global detuning $\Delta(t)$ that acts on every qubit uniformly, and a local detuning $w_{i}\delta(t)$ that acts on individual atoms $i$ with weights $w_{i}$ and time-dependent value $\delta(t)$. The third term is the interaction $V_{ij}=C_{6}/|\vec{x}_{i}-\vec{x}_{j}|^{6}$ between atoms at positions $\vec{x}_{i}$ and $\vec{x}_{j}$ respectively. In systems such as Aquila used in this work, the typical Rabi frequency is $\sim$2.5MHz and the characteristic inter-atomic distance is $\sim 8\mu$m, where up to 256 atoms can be arbitrarily positioned in 2d space.

The goal of hardware implementation is to abstract away the physical Hamiltonian so that it optimally matches the generator of continuous time quantum walks

[TABLE]

where $G(t)=X(t)\hat{\mathcal{G}}+Z_{i}(t)\hat{C}$ is the generator of quantum walks in the independent set subspace spanned by the projector $\mathcal{P}$; $X(t)$ and $Z(t)$ are piecewise constant pulses that recreate the alternating ansatz. The states can be identified with bitstrings by matching $|g\rangle\mapsto|0\rangle$ and $|r\rangle\mapsto|1\rangle$. The Rabi drive can be matched to the $\sigma_{x}$ term by rescaling time $t=2\tau/\Omega$.

A phase jump of the Rabi drive $\phi(t)=-d\phi\Theta(t-t_{0})$ can implement a global phase jump $Z_{i}(t)=d\phi\delta(t-t_{0})$ like that used in state preparation. Local detuning can be used to implement local phase jumps by quickly turning off the Rabi drive, then implementing a fast pulse of local detuning $w_{i}\delta(t)$ on the target atoms such that $d\phi_{i}=w_{i}\int_{0}^{T}\delta(t)dt$. On Aquila, this is most naturally done with a 100 ns triangle pulse with an area of the maximum phase jump, as shown in Fig. 4.

The projection to the independent set subspace can be implemented with a key feature of the Rydberg atom Hamiltonian: the Rydberg blockade. Due to the strong sixth power of the interaction with the distance between atoms, two atoms close together with a distance less than the blockade radius $|\vec{x}_{i}-\vec{x}_{j}|<r_{b}$ will have an energy of the doubly-excited Rydberg state that is much higher than any other scale in the problem. These doubly-excited states can then be integrated out of the dynamics, resulting in an effective subspace of all excitations in the space of independent sets [9, 26]. The graph connectivity is defined by a unit disk graph where the radius is analogous to the blockade radius of the Hamiltonian.

The blockade radius is defined by the Rabi frequency, detuning, and $C_{6}$ coefficient. The dynamic blockade radius, as used for encoding blockaded dynamics, is $r_{d}\equiv(C_{6}/\Omega)^{1/6}$. The static blockade radius, as used for encoding the maximum independent sets into Ising ground states [26], is $r_{s}=(C_{6}/\Delta)^{1/6}$. For Aquila, $\Omega$ has a maximum of $15.8$ rad/$\mu$s and $C_{6}=5420503$ $\mu m^{6}$rad/$\mu$s, so that the dynamic blockade radius is $r_{d}\approx 8.367~\mu$m.

To preserve the independent set condition, adjacent atoms must be close together to ensure that they are well within the blockade radius. However, to preserve independent set dynamics, nearby atoms outside of the unit disk radius must be far enough apart to ensure that the $1/R^{6}$ interaction tail does not spuriously add unwanted interactions. These two factors can be balanced by considering that there is no singular unit disk radius; instead, there is a range of radii that generate the same unit disk graph. The minimum unit disk radius $r_{min}$ is the maximum distance between any two vertices connected by an edge; the maximum unit disk radius $r_{max}$ is the minimum distance between any two vertices not connected by an edge; as an example, consider Fig. 11. To best preserve dynamics, the energy associated with the blockade radius should be much larger (smaller) than the minimum (maximum) distance between atoms

[TABLE]

More specifically, the perturbative Schrieffer-Wolff corrections arising from finite blockade energies must be much smaller than the energy-scale of the Rabi drive. This condition can be maximally satisfied by choosing the blockade radius to be $r_{\text{b}}=\eta\,\sqrt{r_{\text{min}}r_{\text{max}}}$ where $\eta\approx 0.936$ for the atom geometries explored in this work. For more details, see Appendix B and the supplemental of [12].

The space of independent sets of an $N$-vertex nearest-neighbor ring can be represented by an $N$-atom ring of atoms in a 2d atom array. Given the ring where each atom is a distance $D$ from the origin, $r_{\text{min}}=2D\sin(\pi/N)$ and $r_{\text{max}}=2D\sin(2\pi/N)$. The distance is

[TABLE]

In aggregate, it is possible to physically reproduce the state prepared by the phase-walk ansatz (see Eq. 4) by implementing a piecewise-constant, time-dependent evolution with respect to a walk generator

[TABLE]

where $X(\lambda)$ and $Z(\lambda)$ are non-overlapping, piecewise-constant waveforms whose fragments have area $\tau_{q}$ or $\gamma_{q}$, with an equivalent generator of the Rydberg-atom Hamiltonian for a ring of blockaded atoms by matching the independent-set subspace to the Rydberg-blockade subspace and setting the ring radius as above based on the dynamic blockade radius, using the average Rabi-drive amplitude. The global detuning is set to zero throughout. Global phase shifts are implemented via rotating-frame phase shifts of the Rabi drive 555Note that global phase shifts can be done equivalently through a time-integrated global detuning, but this requires a longer physical evolution time that yields a lower fidelity result.. Local phase shifts are implemented via the local detuning term by choosing a waveform $\delta(t)$ such that $\int_{0}^{T}w_{i}\delta(t)dt=\phi_{i}$ in the physical interval $[0,T]$. Global $\sigma_{x}$ accumulation is implemented by choosing a waveform $\Omega(t)$ such that $\int_{0}^{T}\Omega(t)dt=2\tau$ over the minimized physical interval $[0,T]$. The blockade radius and thus the distance between each atom is set by the average of $\Omega(t)$ and thus varies slightly depending on the ansatz. Waveforms and positions are minimally relaxed to conform to slew rate and atom placement constraints inherent to Aquila [61]. An example bloqade program implementing a single walk step is shown in Table 1, and a representative program is shown in Fig. 4.

IV Results: Preparation of product and bracelet states

We present results for the preparation of the product states $z^{\rm half}$ and $z^{\rm MIS}$, together with their corresponding bracelet states $[z^{\rm half}]$ and $[z^{\rm MIS}]$, as described in Section II.4, using the two phase-walk ansätze introduced in Section II.6. These are realized in three scenarios: the “perfect” CTQW dynamics, its approximation by the Rydberg Hamiltonian in noiseless emulation, and implementation on Aquila, sharing the same variational parameters across each case. Our comparison focuses on the success probability and amplification relative to a uniform baseline, contrasting the observed amplification as a function of both the cardinality of the invariant (blockaded) subspace $|\mathcal{V}|$ and the ansatz depth $p$, following the power-law scaling relationships introduced in Section II.5. Particular attention is given to the extent to which the scaling behavior predicted by the ideal CTQW is qualitatively reproduced on noisy hardware.

The unitless walk times $\tau$ are mapped to the minimum feasible physical hardware time given a maximum Rabi frequency of $\Omega=15.8$ rad/$\mu$s. Evolution under the walk generator is implemented with a hardware-minimum rise and fall time of $0.05~\mu$s [61]. Consequently, there are four regimes of Rabi drive pulse length. For walk times $\tau<0.40$, the pulse is triangular with a constant duration of $0.10~\mu$s. Between $0.40\leq\tau<0.59$ the pulse is also triangular, with a duration of $\tfrac{4\tau}{15.8}\,\mu$s, increasing from $0.10$ to $0.15~\mu$s. Between $0.59\leq\tau\leq 0.79$, the optimal pulse is a reduced-amplitude trapezoid of duration $0.15~\mu$s. Beyond $\tau=0.79$, the pulse duration grows linearly as $0.127\,\tau+0.05~\mu$s.

The blockaded subspace is encoded into a ring of atoms arranged in a ring, with nearest-neighbor distances given by Eq. 42. Atom positions are rounded to the nearest $0.1~\mu$m, and the tops and bottoms of the rings are flattened to conform to the $2~\mu$m row spacing constraints of Aquila [61]. Some example atom positions are shown in Fig. 4. All data from Aquila was acquired using QuEra’s exclusive access mode during the month of August 2025, and all shots were post-selected on fully filled arrays.

There is added complexity when using Aquila due to a measurement error of $\sim 7\%$ in misidentification of the Rydberg state as ground and, when using local detuning, an additional $\sim 10\%$ error in misidentification of the ground state, which renders naive frequentist bitstring counting unreliable for large bitstrings. To address this, we employ a Bayesian postprocessing method based on expectation maximization (EM, see Appendix A), which reconstructs the pre-measurement probability distribution by modeling readout as an asymmetric bit-flip channel, with the probability of the target state taken from the reconstructed distribution unless otherwise noted. Uncertainties in the estimated probabilities are quantified using a nonparametric bootstrap to obtain 95% confidence intervals. In Section IV.1, we fit the amplification $A$ to Eq. 13 via weighted nonlinear least squares, taking the 95% CI for $\alpha$ from the fit covariance under the constraint $\alpha<1$. The intervals are propagated through $n=1/(1-\alpha)$, with $n\to\infty$ (reported as “$\geq$”) when the upper bound reaches one.

Figure 5 presents the unprocessed probability distributions for the $z^{\rm half}$ product and bracelet states at $N=7$ across the three considered scenarios, together with the success probabilities obtained from $1000$ shots in noiseless emulation convolved with the asymmetric bit-flip channel, and from the same number of shots on Aquila. Here, and for all other considered target states (see Tables 2 and 3), the reconstructed distributions obtained from the scrambled emulation results agree with the true values within the 95% confidence intervals.

IV.1 Product state preparation

The walk parameters for the target $z^{\rm half}$ and $z^{\rm MIS}$ product states are obtained through ideal simulation of a quantum walk generator, that is, a walk without Rydberg interactions, blockade violations, and finite rise times. They are then directly transferred, with no further optimization, to the time-dependent Rydberg Hamiltonian as described above. The local phase separator (see Eq. 16) is encoded into a minimum-duration triangle pulse with a $50\,$ns rise and fall time commensurate with the constraints of Aquila [61], and a peak value of $\Delta_{\text{max}}=\phi/(50\,\text{ns})\leq 62~\text{rad}~/\mu\text{s}$.

In Fig. 6, the $z^{\rm half}$ targets show higher scaling exponents for both the CTQW reference and noiseless emulation. Via Eq. 14, the CTQW fits correspond to effective polynomial speedup orders $n=\{8.17\,[7.80\!-\!8.58],\;15.3\,[14.6\!-\!16.2],\;53.1\,[48.3\!-\!59.0]\}$, with emulation following closely at $p=1$ and $2$ with $n=\{8.15\,[7.57\!-\!8.82],\;15.1\,[13.8\!-\!16.7]\}$, but diverging at $p=3$ with $n=14.4\,[13.2\!-\!15.7]$. On hardware, the same qualitative behavior is observed, albeit diminished: the scaling exponent increases from $p=1$ ($n=3.70\,[3.21\!-\!4.35]$) to $p=2$ ($n=8.07\,[6.69\!-\!10.1]$) but declines at depth $p=3$ ($n=3.76\,[3.15\!-\!4.66]$).

Figure 7 shows the ratio of success probabilities at $p=2$ relative to $p=1$. Although variability and uncertainty preclude a full fitting-based analysis, the increase in success probability appears to follow a generally rising trend with $N$ which, due to the proportional increase in amplification with $|\mathcal{V}|$, exceeds the convergence gain predicted by an ideal noiseless restricted-depth Grover search [67] for all even $N\geq 16$ and for odd $N=23$.

For $z^{\rm MIS}$ targets, the CTQW fits remain tightly clustered around unity across all depths. At $p=1$, the effective order is $n=247\,[200\!-\!322]$, and at $p=2$ within the limits of numerical precision, the amplification scales directly proportional to $|\mathcal{V}|$ over the range of considered $N$. This can be explained by noting that in $\mathcal{G}$ (here a “Lucas” cube [45]), the subspaces defined by odd and even qubits each induce a hypercube of dimension $\lfloor N/2\rfloor$, containing the antipodal pair $0^{N}\leftrightarrow 1^{N}$ [25], for which the hypercube graph is known to exhibit perfect state transfer [22]. Emulated Rydberg dynamics show similarly strong scaling at depth $p=1$, with $n\geq 261(\alpha\to 1)$, while depth $p=2$ yields a reduced effective order of $n=30.5\,[25.9\!-\!37.3]$. On hardware, amplification follows $n=5.56\,[4.79\!-\!6.63]$ at $p=1$, dropping to $n=3.37\,[2.75\!-\!4.34]$ at $p=2$.

Overall, these results are consistent with the super-linear amplification scaling in $|\mathcal{V}|$ expected of an efficient phase-walk ansatz, and to a lesser extent with increasing depth. The phase separator relies on local detuning, which carries a shot-to-shot per-site coherent error of roughly 10%, leading to deviations from the intended $\pm\pi$ phase shifts that compound across iterations. This is likely the primary reason for the breakdown of increased convergence at higher $p$, in addition to the effects of incoherent errors from longer program times. It is therefore notable that the $z^{\rm half}$ targets still show improved convergence at $p=2$. For the $z^{\rm MIS}$ targets, the CTQW dynamics predict convergence near unity across the range of $N$. Consequently, the decrease in amplification from $p=1$ to $p=2$ in both emulation and on-hardware Rydberg dynamics is consistent with saturation of CTQW-based convergence at $p=1$, where further iterations offer little additional amplification and are outweighed by the loss in success probability from accumulated phase errors.

IV.2 Bracelet state preparation

In bracelet state preparation, direct parameter transfer from the ideal CTQW-based ansatz to its approximation via the Rydberg Hamiltonian is unreliable due to the effect of van der Waals interactions over $\tau_{\rm{eff}}$, which is significantly longer than the $\tau$ used in product state preparation. These interactions introduce an additional positive phase through Rydberg-Rydberg couplings, contributing coherent phase error to the walk evolution and breaking the $-\gamma\equiv\gamma$ equivalence that holds for the ideal CTQW. We mitigate this effect through joint optimization of $\tau_{\rm{eff}}$ and $\bm{\gamma}$, taking as the objective the mean success probability

[TABLE]

to identify parameter sets that achieve high convergence simultaneously in both cases, with the global phase separator (see Eq. 33) encoded into a piecewise constant phase profile of the Rabi drive.

Compared to the CTQW-derived parameters, the jointly optimal parameters systematically require longer walk times. In Fig. 8, the ratio $\tau_{\rm{eff}}^{\rm{Ryd}}/\tau_{\rm eff}^{\rm CTQW}$ reveals that target states with smaller spectral gaps require the largest inflation in $\tau_{\rm eff}$, up to approximately $10\times$. More resolvable targets cluster around a factor of $1.5$ to $2.5$. This behavior indicates that van-der-Waals-induced phases compress the effective gap, requiring longer walk times to resolve the target.

Figure 9 presents the amplification and fidelity scaling for bracelet state preparation, comparing ideal CTQW dynamics, noiseless Rydberg emulation, and Aquila hardware. Unlike the product state demonstrations, which sweep ansatz depth at fixed $\tau_{\rm eff}\approx\pi/2$, bracelet state preparation is performed at a single optimized $(\tau_{\rm eff},p)$ pair for each target, with $\tau_{\rm eff}$ ranging from $\sim\!5$ to $\sim\!20$ across targets (see Table 3 of Appendix C). The top panels show amplification (Eq. 12) as a function of graph size $|\mathcal{V}|$, with data split by the parity of $N$. For $\left[z^{\rm MIS}\right]$ (top right), power-law fits (Eq. 13) to the hardware data yield $\alpha=0.84\pm 0.07$ for even-$N$ instances and $\alpha=0.69\pm 0.08$ for odd-$N$, both indicating super-quadratic ($\alpha>0.5$) scaling. The even-$N$ instances correspond to a two-fold degenerate orbit ($|[z]|=2$) and exhibit near-linear scaling across all three cases, while odd-$N$ instances span an $N$-fold superposition and show somewhat reduced but still clear power-law scaling. For $\left[z^{\rm half}\right]$ (top left), noiseless emulation follows the CTQW prediction, but hardware results show no clear power-law trend ($\alpha=0.18\pm 0.18$), most likely due to the combined effect of longer walk times and greater ansatz depth.

As shown in the bottom panels of Fig. 9, $\left[z^{\rm half}\right]$ targets require systematically longer walk times than $\left[z^{\rm MIS}\right]$ targets, which is consistent with the $\left[z^{\rm MIS}\right]$ targets having larger spectral gaps (bottom right). The relative fidelity $P_{\rm Hardware}/P_{\rm Emulation}$ (bottom-left) decays approximately exponentially with the cumulative walk time and, fitting to the form $\exp(-t/T_{2}^{*})$ where $t=2\tau_{\rm eff}/\Omega$, yields an effective dephasing time $T_{2}^{*}\approx 1.07\,\mu$s. Together, these observations indicate that $\left[z^{\rm MIS}\right]$ maintains power-law scaling because its larger spectral gaps permit preparation within the coherence window, while $\left[z^{\rm half}\right]$ targets require walk times that exceed $T_{2}^{*}$ and consequently lose their scaling advantage.

The fitted $T_{2}^{*}$ is notably shorter than the single-qubit Rabi coherence time of $\sim\!5\,\mu$s [61]. This reduction likely reflects the accumulation of phase errors across the many Rabi phase jumps. Because the ansatz depth scales with the required walk time (see Eq. 35), longer $\tau_{\rm eff}$ entails proportionally more phase jumps ($p=10$–$48$ across the targets studied here). Future work could explore whether fidelity on hardware might be improved by using fewer, longer walk segments.

Despite these limitations, the hardware results are consistent with the non-adiabatic $\tau_{\rm eff}\propto 1/\Delta_{\rm min}$ scaling predicted by the frequency-resolution model (Eq. 37). As shown in the bottom-right panel of Fig. 9, the optimized walk times cluster around the $\tau\propto 1/\Delta_{\rm min}$ reference line rather than the $\tau\propto 1/\Delta_{\rm min}^{2}$ scaling that would arise from adiabatic preparation [1]. While the van-der-Waals-induced phase compression increases the proportionality constant relative to the ideal CTQW (Fig. 8), the scaling relationship is preserved, confirming that the phase-walk ansatz successfully implements the non-adiabatic “shortcut” on current hardware.

V Results: Coherent quenches of bracelet states

While the projected probability distribution $P(z)=|\langle z|\psi\rangle|^{2}$ is a reasonable proxy to the fidelity of a state, it is incomplete. Notably, measurements in the Z basis ignore all phase information in the wavefunction, which makes the translationally averaged incoherent mixture

[TABLE]

and the coherent bracelet state

[TABLE]

have identical populations ($|\langle z|\rho_{\rm inc}|z\rangle|^{2}=|\langle z|\rho_{\rm coh}|z\rangle|^{2}$), even though there are nontrivial off-diagonal matrix elements in $\rho_{\rm coh}$ and (if prepared exactly) none in $\rho_{\rm inc}$.

We implement quenches from $|[z^{\rm half}]\rangle$ and, for comparison, from the representative product state $|z^{\rm half}\rangle$, using the resulting dynamics as an indicator of bracelet-state preparation, as the post-quench evolution of the coherent bracelet state diverges from that of its incoherent counterpart. The two states are prepared as shown in Fig. 5 and the protocol extended by adding an extra Rabi pulse of duration $t=2\tau/\Omega$.

Figure 10 illustrates that, at short evolution times, the dynamics of these two states are equivalent, but at longer times, interference effects cause the distributions to diverge. The coherent and incoherent perfect walk dynamics (gray dashed) show the earliest divergence and clearly distinguish the coherent and incoherent states. Emulated neutral atom dynamics (purple) follow the perfect walk dynamics, as expected from parameter matching the virtual and physical systems. However, at longer times the similarity diverges due to perturbative blockade violations and $R^{-6}$ Rydberg interactions outside of the blockade radius (see Appendix B). This suggests that, while for short times, such as state preparation, the two dynamics approximately match. For longer evolution times, the dynamic blockade radius approximation begins to break down, and the Rydberg atom Hamiltonian can no longer accurately represent CTQW on independent set subspaces.

We find that results from Aquila qualitatively track the dynamics of the noiseless emulation. The most prominent difference in the observed dynamics being the higher population at $\tau\approx 5.8$ in the $|[z^{\rm half}]\rangle$ quench (lower panel), which is consistent with the predicted dynamics of the coherent bracelet state. However, there is notable divergence from the noiseless prediction. Most clearly, in the quench from $|[z^{\rm half}]\rangle$ between $\tau\approx 1.9$ and $3.2$, two closely spaced peaks merge into a single broader maximum, consistent with the accumulation of dephasing error. For the incoherent quench, the 95% CI overlaps the ideal curve for most $\tau$, but it has a much broader interval relative to the coherent case – which could result from use of local detuning during the initial preparation stage (as discussed in Section IV.1). Overall, although the on-hardware timescales ($\leq 2~\mu$s) lie within the Rabi-drive coherence time ($\approx 5~\mu$s), the deviations are not inconsistent with accumulated incoherent error over this window. We leave a deeper investigation of the specific contributing factors to future work.

VI Conclusion

This work has presented an implementation and characterization of continuous-time quantum walk (CTQW) based ansätze on neutral atom hardware. By transferring the phase-walk structure from abstract models to execution under the Rydberg Hamiltonian, we established a direct link between ideal CTQW dynamics and their realization on a noisy intermediate-scale quantum (NISQ) device. Using known target states as controlled testbeds, we probed how constructive interference, amplification, and entanglement emerge in constrained Hilbert spaces subject to realistic hardware limitations. Variational optimization and error-mitigation were carried out using Pawsey’s “Setonix” and “Ella” supercomputing clusters, while hardware execution was performed on QuEra Computing’s cloud-accessible, analog-mode neutral-atom processor Aquila.

Unlike earlier approaches [30] that used quantum walks as variational routines for preparing unknown states, here we target nontrivial but efficiently describable states [33] chosen as benchmarks for assessing interference and amplification on NISQ hardware. In particular, we focus on bracelet states that respect the dihedral symmetry of an $N$-vertex ring, and product states whose preparation nonetheless requires nontrivial dynamics through the constrained Hilbert space. For the bracelet states, entanglement emerges from the non-separable structure of the independent-set Hilbert space itself. We find that this constrained space, together with restricted local controls, is sufficient to support variational ansätze capable of preparing such structured target states.

We demonstrate a key use of quantum processors as analogue simulators of abstract models: the dynamics of CTQWs on independent-set graphs can be mapped directly onto the nonequilibrium dynamics of the Rydberg Hamiltonian. The independent-set constraint is naturally enforced by the Rydberg blockade, enabling a hardware representation of nonequilibrium dynamics on constrained subspaces that has previously only been explored in the context of quantum scar dynamics [9]. Despite substantial measurement errors, we find that Aquila can reproduce the same target states as the abstract CTQW system through direct parameter transfer with little to no further optimization.

For product-state targets, hardware circuits reproduce the hallmark amplification through constructive interference of efficient CTQW-inspired protocols. The $z^{\rm MIS}$ states benefit from perfect state transfer on an embedded hypercube subgraph, with ideal CTQW dynamics predicting near-unity amplification even at a single ansatz iteration. The $z^{\rm half}$ states provide a more representative benchmark. Here, our results suggest a conservative theoretical upper-bound on the achievable polynomial-order speedup in query complexity of $n\approx 8$ at ansatz depth one, rising to $n\approx 15$ at depth two and $n>50$ at depth three. On Aquila, we observe $n\approx 4$ at depth one and $n\approx 8$ at depth two, demonstrating that the fast-forward dynamics characteristic of efficient CTQW protocols are already accessible at low circuit depth. This opens the possibility of realizing super-quadratic amplification in algorithmic applications on current hardware.

For bracelet-state targets, our results extend the non-adiabatic shortcut demonstrated for product-state MIS preparation via sweep-quench-sweep protocols [39] to the preparation of symmetric entangled states, successfully applying a CTQW-based framework that predicts an advantage over adiabatic protocols. While the influence of van der Waals interactions in the Rydberg dynamics precluded the direct transfer of CTQW-derived parameters, we found that it was nevertheless possible to identify parameters that are jointly optimal for both the ideal CTQW and the Rydberg evolution, with resulting walk times retaining the $\tau_{\rm eff}\propto 1/\Delta_{\rm min}$ scaling predicted by the non-adiabatic protocol.

While this work is an initial step in implementing CTQW on neutral atom hardware, there are many future directions to explore. Instead of directly optimizing to some efficiently describable target state, future work could instead use the quantum walks framework to prepare resource states for hybrid optimization [64] or machine learning tasks [17]. Furthermore, this work focused on the walk graph formed by independent sets of an $N$-vertex ring; future work could implement different walk graphs induced by other independent set graph constraints or Hamiltonian engineering techniques [20] to explore quantum walks on tailored subspaces.

Acknowledgements

This work was supported by resources provided by the Pawsey Supercomputing Research Centre’s Setonix Supercomputer [46], with funding from the Australian Government and the Government of Western Australia. This work and the Pawsey Supercomputing Research Centre’s Quantum Supercomputing Innovation Hub were made possible by a grant from the Australian Government through the National Collaborative Research Infrastructure Strategy (NCRIS). We thank Sheng-Tao Wang and the rest of the theory team at QuEra Computing for their support. PJE and EM thank the entire Quantum Supercomputing Innovation Hub team at Pawsey and QuEra.

Data Availability

Data and software supporting the results reported in this work, including source code and datasets obtained from QuEra’s Aquila system, are publicly available [43].

Appendix A Bayesian inference for distribution reconstruction

A central challenge in analyzing coherent state preparation is measurement error, where a [math] is mis-detected as a $1$ or vice versa. This appendix details our use of the expectation maximization (EM) algorithm, a Bayesian inference procedure, to reconstruct the underlying distribution from measured bitstring data. Our approach is related to recent iterative methods for error mitigation [48], which employ histogram-based unfolding. Here, we instead construct a computationally tractable model by parameterizing only the blockaded subspace $\mathcal{V}$ directly and regularizing contributions from its complement.

A.1 Bayesian inference

Bayesian inference provides a framework to address uncertainties arising through noisy measurement [40, 11]. Given observed data $Z$ and model parameters $\bm{\phi}$, Bayes’ rule states

[TABLE]

where the prior distribution $P(\bm{\phi})$ is the model before data collection, $P(Z\mid\bm{\phi})$ is the likelihood of observing $Z$ given parameters $\phi$, and the posterior distribution $P(\bm{\phi}\mid Z)$ describes how plausible the values of $\bm{\phi}$ are given $Z$. Obtaining parameters that best explain the observed data then occurs through maximization of the likelihood.

A.2 Likelihood under measurement error

Measurement error on Aquila is well described by an asymmetric bit-flip channel acting independently on each qubit [61]. The probability of recording a bitstring $z$ given pre-measurement string $s$ is

[TABLE]

where $j$ indexes the bits of $z$ and $s$, which essentially convolves the pre-measurement probabilities with the transition matrix

[TABLE]

where $P_{ab}$ is the probability for the transition $a\rightarrow b$, $P_{01}=1-P_{00}$ and $P_{10}=1-P_{11}$.

As our aim is to construct the pre-measurement distribution over the subspace of graph vertices $\mathcal{V}$, we consider a model with parameters $\bm{\phi}_{|\mathcal{V}|}=\{\phi^{k}\}_{k=1}^{|\mathcal{V}|}$ for each possible originating bitstring $s^{k}\in\mathcal{V}$, and a reduced set of parameters $\bm{\phi}^{\perp}=\{\phi^{\perp}_{j}\}_{j=1}^{N}$ for bitstrings $s\in\mathcal{V}^{\perp}=\{0,1\}^{N}\setminus\mathcal{V}$. Here each $\phi^{\perp}_{j}$ models the probability that bit $j$ of an observed bitstring is $1$ in $\mathcal{V}^{\perp}$, using independent Bernoulli parameters,

[TABLE]

a choice informed by our assumption of independent single-qubit noise channels. The full parameter set is

[TABLE]

Combining the contributions from $\mathcal{V}$ and $\mathcal{V}^{\perp}$, the model probability of measuring bitstring $z^{i}$ is

[TABLE]

where $L_{k,i}=K(z^{i}\mid s^{k})$ is the likelihood for $s^{k}\in\mathcal{V}$, and the likelihood for $s\in\mathcal{V}^{\perp}$ is

[TABLE]

A.3 Expectation Maximization

The expectation–maximization (EM) algorithm is an iterative method for likelihood maximization. The algorithm alternates between two steps, which are guaranteed to monotonically converge to a local optimum of the likelihood [24]. The first is an “E-step” that computes responsibilities – the posterior probability that each component of the model generated observation $z^{i}$:

[TABLE]

with $\sum_{k=1}^{|\mathcal{V}|}\rho^{(t)}_{k,i}+\rho_{\perp,i}^{(t)}=1$

The second is the “M-step”, which updates the parameters using the average of the responsibilities over all observations, which for $s^{k}\in\mathcal{V}$ is:

[TABLE]

where $|Z|$ is the number of observed bitstrings. Updating $\bm{\phi}^{\perp}$ requires estimation of the expected pre-measurement bit values. For bit $j$ this is given by:

[TABLE]

where $T_{j}$ denotes the true pre-measurement value of bit $j$, and $\rho^{(t)}_{\perp,i}$ is the responsibility of $\mathcal{V}^{\perp}$ for $z^{i}$ at iteration $t$. The strength of the Rydberg blockade effect leads to a small contribution from $\mathcal{V}^{\perp}$, so $\alpha$ and $\beta$ are introduced as regularizing terms (a $\mathrm{Beta}(\alpha,\beta)$ prior [40]) on each $\phi^{\perp}_{j}$ to prevent the estimate from collapsing to exactly [math] or $1$. The expectation value $\mathbb{E}[T_{j}\mid z^{i}_{j}]$ is computed by inversion of the bit-flip channel (Eq. 47):

[TABLE]

In this work, we choose an “uninformative” prior, initializing $\bm{\phi}_{\mathcal{V}}$ as a uniform distribution and $\bm{\phi}^{\perp}$ to a uniform baseline with an uniform $\mathrm{Beta}(1,1)$ prior [11]. Parameters for the bit-flip channel are $(P_{00},P_{11})=(0.99,0.93)$ under standard operation [61] and $(P_{00},P_{11})=(0.90,0.93)$ when using local detuning. We use the convergence criterion

[TABLE]

where we set $\epsilon=10^{-8}$. Uncertainty in the estimated probability of measuring a given target $z^{*}$, taken from the optimized $\bm{\phi}$, is computed using a nonparametric bootstrap to a $95\%$ confidence interval. We choose this approach as it adapts well to model complexity and finite-sample variability with minimal assumptions about the underlying distribution [27].

Appendix B Blockade subspace guarantees from perturbation theory

This appendix details the derivation of the scaling of the positions to best match the blockade radius $r_{b}$, which may be derived from perturbation theory. In the limit of large interaction strength between atoms within the blockade radius, we may derive the first-order correction and perturbative Hamiltonian from a virtual excitation to the blockaded space. By balancing the contribution from these first-order corrections and long-range $r^{6}$ interactions outside of the blockade radius, the error Hamiltonian is minimized, maximizing the effective projective Hamiltonian evolution.

For each edge between vertices in the unit disk graph, consider the 2-qubit subspace

[TABLE]

where the labels represent Hamiltonian terms. Given a definition of Pauli matrices $\sigma_{x}\equiv|g\rangle\langle r|+|r\rangle\langle g|$ and $n\equiv|r\rangle\langle r|$, the effective error Hamiltonian is [12]

[TABLE]

The first term is the sum over the perturbative corrections from a virtual excitation to the doubly excited subspace, which results in a hopping term $|gr\rangle\leftrightarrow|rg\rangle$ and Stark shift. The second term is the sum over all van der Waals interactions outside the blockade radius.

Minimizing the contribution of the error Hamiltonian is equivalent to minimizing its norm

[TABLE]

Defining the Blockade radius $|\Omega|\equiv C_{6}/r_{b}^{6}$ and assume that the contribution from blockaded edges comes from $n_{b}$ edges per vertex at a maximum distance $r_{\text{max}}$, and the contribution from unblockaded edges comes from $n_{u}$ edges per vertex at a minimum distance of $r_{\text{min}}$; $n_{b}$ and $n_{u}$ can be scaled to include more weakly interacting edges by rescaling the sum. Then, the norm of the error is

[TABLE]

Given a scaling $\lambda$ between virtual coordinates $\overline{x}$ and physical atom positions $x$ as $x=\lambda\overline{x}$, the error can be minimized by minimizing with respect to $\lambda$ as $\partial_{\lambda}||H||=0$ to find

[TABLE]

Although the prefactor scales weakly with the $1/24$th power, as shown in Fig. 12, we find that its inclusion tangibly improves the CTQW fidelity at small $\tau$. For the 1d chain, $n_{b}=2$ and $n_{u}\approx 2.220$, so the prefactor shifts the blockade radius by a factor of $0.939$.

For a 2d King’s lattice, $n_{b}=2.25$ and $n_{u}\approx 7.791$, so the prefactor shifts the blockade radius by a factor of $0.940$. The magnitude of the minimized error term is

[TABLE]

Reducing the ratio between the maximum distance between vertices connected by a unit disk edge, and the minimum distance between vertices not connected by a unit disk edge will reduce the error. For the 1d chain, $r_{\text{max}}/r_{\text{min}}=1/2$, so $||H_{\text{err}}||/N\approx(0.181\Omega)^{2}$. For the 2d king’s graph, $r_{\text{max}}/r_{\text{min}}=1/\sqrt{2}$, so $||H_{\text{err}}||/N\approx(0.723\Omega)^{2}$.

Appendix C State Preparation Results

This appendix collects detailed numerical results for state preparation demonstrations. Table 4 reports the power-law fit parameters ($c$, $\alpha$, and $R^{2}$) for the amplification scaling $A=c\,|\mathcal{V}|^{\alpha}$ shown in Fig. 6. Tables 2 and 3 provide per-instance success probabilities for product states and bracelet states, respectively.

In both Tables 2 and 3, the State column identifies the target configuration, where $N$ denotes the number of vertices in the constraint graph and $|\mathcal{V}|$ the number of vertices in the corresponding walk graph $\mathcal{G}$. The Depth ($p$) column records the number of ansatz layers. For product state preparation (Table 2), the variationally optimised parameters are the fiducial and walk times $\tau_{0}$ and $\tau_{1}$, and $J_{\rm eff}$ is the effective coupling (see Eq. 28) computed from the walk times. For bracelet states (Table 3), the relevant parameters are the effective evolution time $\tau_{\text{eff}}=\tau p$ and the phase jumps $\gamma$.

The Naïve count column gives the fraction of bitstrings matching the target state directly from raw shot data. The Perfect column reports the corresponding preparation probability under ideal CTQW dynamics, while the Emulation column shows the same quantity obtained from noiseless Rydberg-atom emulation. Hardware performance is reported in the EM (H) column, which shows the success probability obtained from demonstration on Aquila after measurement-error mitigation (see Appendix A), together with the $95\%$ CI. As a check for consistency, the EM (E) column presents values obtained from $1000$ emulated shots convolved with the noisy measurement channel.