Computationally Tractable Offline Quantum Experimental Design for Nuclear Spin Detection

TL;DR

This paper introduces a deep learning-based method with a surrogate information gain metric to optimize measurement data selection, significantly reducing experimental time in nuclear spin detection for quantum technologies.

Contribution

It extends the SALI model with a new variance-based optimization approach, validated on experimental and simulated data, to improve efficiency in nuclear spin environment characterization.

Findings

85% reduction in measurement time with experimental data

60% reduction in total experiment time with simulated data

First validation of SALI on experimental data

Abstract

The characterization of nuclear spin environments in solid-state devices plays an important role in advancing quantum technologies, yet traditional methods often demand long measurement times. To address this challenge, we extend our recently developed deep-learning-based SALI model (Signal-to-image ArtificiaL Intelligence) by introducing the surrogate information gain (SIG) to optimize the selection of data points in the measurements. This approach significantly reduces time requirements in experiments while preserving accuracy in nuclear spin detection. The SIG is a figure of merit based on the expected variance of the signal, which is more straightforward to compute than the expected information gain rooted in Bayesian estimation. We demonstrate our approach on a nitrogen-vacancy (NV) center in diamond coupled to $^{13}$C nuclei. In the high-field regime, our variance-based optimization is validated with experimental data, resulting in an 85$\%$ reduction in measurement time for a modest reduction in performance. This work also constitutes the first validation of SALI on experimental data. In the low-field regime, we explore its performance on simulated data, predicting a 60$\%$ reduction in the total experiment time by improving the temporal resolution of the measurements and applying SIG. This demonstrates the potential of integrating deep learning with optimized signal selection to enhance the efficiency of quantum sensing and nuclear spin characterization, paving the way for scaling these techniques to larger nuclear spin systems.

Tables

Table 1. Table S1: Numerical values of the predicted coupling constants for the spins highlighted by green boxes in Fig. 1 (a). These predictions match those found in Supplementary Table 2 of Letter [ 3 ] .

	2 inputs		4 inputs		SIG
Nucleus	$A^z$ (kHz)	$A^{\perp }$ (kHz)	$A^z$ (kHz)	$A^{\perp }$ (kHz)	$A^z$ (kHz)	$A^{\perp }$ (kHz)
C1	-	-	-	-	-	-
C2	-24.32(0)	22.07(1)	-24.28(0)	22.11(1)	-24.33(1)	22.10(2)
C3	-20.64(0)	41.54(1)	-20.66(0)	41.66(3)	-20.63(1)	41.50(2)
C4	-17.75(1)	8.73(2)	-17.71(1)	9.00(2)	-17.64(1)	8.84(5)
C5	-14.8(1)	4.7(7)	-14.67(1)	9.29(1)	-14.56(1)	10.40(5)
C6	-7.96(1)	15.60(3)	-7.93(1)	5.11(4)	-7.97(2)	19.30(3)
C7	-8.07(0)	20.73(1)	-8.14(1)	21.30(2)	-8.02(1)	20.60(4)
C8	-7.86(1)	6.05(3)	-0.86(2)	6.48(4)	-5.28(1)	12.80(4)
C9	-4.79(1)	2.83(2)	-4.82(1)	3.59(1)	-5.19(3)	2.75(3)
C10	-2.93(0)	10.13(1)	-2.79(1)	7.93(2)	-3.05(0)	12.60(3)
C11 - C16	-2.54(0)	12.60(1)	-1.09(3)	12.80(5)	-	-
C17 - C22	-1.04(0)	12.96(1)	-1.13(1)	13.6(1)	-	-
C23 - C27	1.94(0)	13.11(1)	1.93(0)	13.19(1)	-	-
C28 - C29	1.95(0)	13.06(1)	2.29(2)	4.79(4)	13.85(1)	8.9(1)
C30	2.53(3)	9.55(1)	3.51(1)	5.03(1)	14.01(3)	6.3(1)
C31-32	4.03(1)	8.15(2)	4.30(0)	7.29(2)	4.19(1)	8.11(3)
C33	4.07(1)	9.19(1)	-	-	3.77(3)	10.80(4)
C34	4.26(0)	12.16(4)	-	-	3.90(3)	12.6(2)
C35	7.04(1)	2.13(1)	7.01(0)	2.61(2)	7.00(1)	2.27(1)
C36	-	-	-	-	14.1(1)	5.5(1)
C37	9.19(1)	5.11(4)	9.17(1)	5.14(3)	9.24(1)	5.16(3)
C38	13.82(0)	7.97(6)	13.99(0)	8.7(1)	13.80(1)	7.90(3)
C39*	-	-	14.06(2)	6.85(2)	13.73(2)	16.23(8)
C40	11.38(0)	58.82(1)	11.29(0)	58.75(2)	11.37(1)	59.17(2)
C41 - C42	19.84(1)	5.71(2)	20.01(1)	5.96(1)	19.84(1)	5.98(3)
C43*	-	-	23.18(2)	6.70(6)	-	-
C44*	27.87(5)	2.91(7)	28.55(3)	3.03(3)	-	-
C45	32.86(5)	3.00(6)	31.46(5)	3.26(7)	32.38(3)	2.59(4)
C46*	44.30(2)	3.81(5)	44.19(1)	3.65(3)	48.28(2)	12.23(4)
C47	36.23(1)	26.44(3)	36.22(1)	26.48(2)	36.36(1)	26.55(2)
C48	48.33(1)	11.12(3)	48.50(0)	10.17(2)	48.36(1)	11.03(6)
Total:	27		27		25

Table 2. Table S2: Numerical values of the predicted coupling constants for the spins highlighted by red boxes in Fig. 1 (a). These predictions do not match those found in Supplementary Table 2 of Letter [ 3 ] . The strongly coupled nuclear spin (C1 in Table S1 ) reported in that work lies outside our sampling range and is therefore not detected in our predictions. Instead, our model identifies these other spins within our sampling range that may be influenced by C1.

	2 inputs		4 inputs		SIG
Nucleus	$A^z$ (kHz)	$A^{\perp }$ (kHz)	$A^z$ (kHz)	$A^{\perp }$ (kHz)	$A^z$ (kHz)	$A^{\perp }$ (kHz)
C1E	-43.20(4)	2.71(3)	-42.78(2)	3.24(2)	-42.91(3)	2.78(3)
C2E	-	-	-30.93(6)	2.43(4)	-30.12(2)	2.59(1)
C3E	-26.31(2)	3.14(4)	-26.91(6)	3.3(1)	-	-
Total:	2		3		2

Table 3. Table S3: Numerical results from the left plots of Fig. 1 (b) and Fig. 3 (a) in the main text.

	F1-score: high-field regime
Nuclear spins	11 hours	8 hours	4 hours	3.2 hours	2.4 hours	1.6 hours
1	0.6486	0.8638	0.9579	0.9476	0.9292	0.8951
2	0.8997	0.9583	0.9695	0.9690	0.9676	0.9625
3	0.9724	0.9820	0.9685	0.9682	0.9677	0.9658
4	0.9899	0.9869	0.9663	0.9659	0.9654	0.9634
5	0.9920	0.9867	0.9629	0.9625	0.9617	0.9596
6	0.9920	0.9849	0.9612	0.9609	0.9598	0.9572
7	0.9912	0.9830	0.9578	0.9571	0.9561	0.9527
8	0.9899	0.9810	0.9543	0.9537	0.9523	0.9484
9	0.9882	0.9785	0.9520	0.9514	0.9494	0.9452
10	0.9875	0.9766	0.9481	0.9474	0.9452	0.9403
11	0.9859	0.9735	0.9446	0.9438	0.9414	0.9356
12	0.9845	0.9711	0.9414	0.9401	0.9376	0.9315
13	0.9833	0.9692	0.9381	0.9366	0.9338	0.9267
14	0.9810	0.9654	0.9338	0.9321	0.9289	0.9207
15	0.9797	0.9626	0.9304	0.9288	0.9248	0.9157
16	0.9780	0.9598	0.9262	0.9236	0.9198	0.9096
17	0.9760	0.9571	0.9226	0.9204	0.9159	0.9049
18	0.9743	0.9538	0.9180	0.9154	0.9104	0.8978
19	0.9726	0.9506	0.9142	0.9115	0.9062	0.8926
20	0.9705	0.9475	0.9102	0.9074	0.9012	0.8865
21	0.9690	0.9439	0.9057	0.9024	0.8962	0.8799
22	0.9665	0.9405	0.9009	0.8976	0.8906	0.8738
23	0.9642	0.9373	0.8961	0.8926	0.8850	0.8667
24	0.9620	0.9332	0.8917	0.8878	0.8796	0.8596
25	0.9596	0.9296	0.8869	0.8829	0.8738	0.8519
26	0.9567	0.9257	0.8816	0.8769	0.8675	0.8440
27	0.9544	0.9216	0.8764	0.8719	0.8619	0.8360
28	0.9519	0.9172	0.8714	0.8665	0.8553	0.8277
29	0.9493	0.9131	0.8658	0.8598	0.8487	0.8195
30	0.9463	0.9086	0.8602	0.8546	0.8416	0.8111
31	0.9428	0.9039	0.8545	0.8480	0.8349	0.8020
32	0.9404	0.8999	0.8491	0.8419	0.8284	0.7926
33	0.9371	0.8950	0.8433	0.8357	0.8214	0.7842
34	0.9340	0.8903	0.8370	0.8289	0.8131	0.7737
35	0.9304	0.8851	0.8312	0.8228	0.8058	0.7639
36	0.9276	0.8805	0.8258	0.8170	0.7985	0.7550
37	0.9237	0.8756	0.8197	0.8102	0.7915	0.7458
38	0.9205	0.8702	0.8127	0.8033	0.7838	0.7350
39	0.9172	0.8649	0.8073	0.7969	0.7759	0.7257
40	0.9130	0.8593	0.7993	0.7890	0.7669	0.7139
41	0.9088	0.8538	0.7937	0.7827	0.7590	0.7037
42	0.9045	0.8485	0.7868	0.7748	0.7511	0.6928
43	0.9009	0.8425	0.7800	0.7672	0.7420	0.6824
44	0.8965	0.8364	0.7727	0.7592	0.7339	0.6730
45	0.8922	0.8300	0.7651	0.7516	0.7250	0.6610
46	0.8887	0.8248	0.7585	0.7442	0.7164	0.6510
47	0.8837	0.8182	0.7506	0.7363	0.7069	0.6396
48	0.8787	0.8111	0.7425	0.7275	0.6971	0.6276
49	0.8747	0.8052	0.7350	0.7190	0.6882	0.6174
50	0.8694	0.7983	0.7268	0.7107	0.6783	0.6057

Table 4. Table S4: Numerical results from the middle plots of Fig. 1 (b) and Fig. 3 (a) in the main text.

	MAE of $A^z$ (Hz): high-field regime
Nuclear spins	11 hours	8 hours	4 hours	3.2 hours	2.4 hours	1.6 hours
1	37.3579	55.9494	34.1342	34.4847	35.1491	36.7516
2	30.5994	36.5999	32.3540	32.5773	32.7347	33.5186
3	29.6355	31.6862	34.2294	33.9646	34.2671	35.0037
4	30.4830	31.5786	35.6287	35.6959	36.2522	36.7835
5	32.2229	33.2231	37.7326	37.9817	38.2145	39.2471
6	33.3882	34.8250	39.4208	39.6920	39.9132	41.0156
7	35.0246	36.8593	41.4483	41.6264	42.1264	43.0197
8	36.9964	38.9334	43.0666	43.3798	44.0352	44.9285
9	38.4091	40.7201	45.0028	45.5925	46.1187	47.2475
10	40.0213	42.5951	46.9020	47.5271	47.8071	49.2318
11	41.8209	44.6581	48.9746	49.6040	49.9161	51.6924
12	43.0590	46.2422	50.6468	50.9285	51.8300	53.2450
13	44.7005	48.1559	52.3590	52.9511	53.7095	55.3132
14	46.4185	50.0166	54.2082	54.9139	55.6775	57.2206
15	47.8875	51.8365	56.3029	56.6690	57.5752	59.4111
16	49.9397	53.8908	58.4786	58.7619	59.6359	61.2832
17	51.0567	55.3414	59.7513	60.1687	61.3133	63.0093
18	52.7545	57.2409	61.5724	62.2434	63.1865	65.0949
19	54.0236	59.0522	63.5470	64.2396	65.0678	66.7282
20	55.8557	60.8222	65.5818	66.0567	66.9224	69.1059
21	57.5917	62.6874	67.3422	67.8250	68.8935	71.2285
22	59.2823	64.6295	69.4162	70.0028	71.2196	73.4623
23	60.4226	66.4661	71.2642	72.0325	72.9794	75.2506
24	62.6901	68.5863	73.3250	74.0075	75.1410	77.5501
25	64.4751	70.7731	75.7269	76.2446	77.5201	79.8323
26	66.0593	72.5829	77.8761	78.4538	79.6114	81.6559
27	67.5344	74.4330	79.2851	80.1474	81.0705	83.4549
28	69.6870	76.6909	81.6278	82.3435	83.4338	85.9400
29	71.4118	78.6176	83.9669	84.7580	85.7953	87.9928
30	73.4251	80.8052	86.3702	86.9349	88.1992	90.5276
31	75.5411	83.0562	88.3005	89.4740	90.3810	92.5552
32	77.2835	84.9648	90.8000	91.2820	92.4126	95.2764
33	79.4161	87.2626	93.1314	93.9148	95.0340	97.2793
34	81.5946	89.7932	95.5567	95.9737	97.3636	99.8144
35	83.5816	91.9698	97.9221	98.9156	99.9614	102.1157
36	85.8841	93.7312	100.4024	101.1566	101.9878	104.5278
37	87.9473	96.1841	103.3925	103.5926	104.4820	107.1818
38	90.1387	98.5522	105.8394	106.3856	107.4910	110.0982
39	92.4781	101.1007	108.2088	108.9615	109.9228	112.9811
40	94.7375	103.7605	111.3539	112.3530	112.5630	115.6889
41	97.2296	106.7776	114.6005	114.8631	116.2304	118.9873
42	99.4325	108.7256	116.5325	117.1347	118.7316	121.0262
43	102.1366	111.5452	119.7913	120.5075	121.1770	124.3705
44	104.2050	114.0404	122.7284	123.2835	123.7517	127.0794
45	107.0253	116.8003	125.2068	126.1152	127.1110	129.4451
46	109.4121	119.3892	128.4924	129.0681	130.4413	133.5998
47	112.2885	122.0577	131.2663	131.8696	133.2541	136.4695
48	114.3209	124.9334	134.5305	135.3156	136.1304	139.7813
49	116.9255	127.3942	137.2424	137.4102	138.9628	143.2425
50	119.7766	130.8862	140.3613	141.2116	142.8719	145.9034

Table 5. Table S5: Numerical results from the right plots of Fig. 1 (b) and Fig. 3 (a) in the main text.

	MAE of $A^{\perp }$ (Hz): high-field regime
Nuclear spins	11 hours	8 hours	4 hours	3.2 hours	2.4 hours	1.6 hours
1	92.7411	179.0140	99.5267	101.9721	105.8136	111.5539
2	82.9324	118.2944	96.1301	98.1266	101.8063	109.6251
3	84.3050	103.4392	103.1559	106.2852	109.4395	116.2342
4	87.9073	105.2502	111.9473	114.4902	118.2498	124.2189
5	94.2537	113.0375	121.6607	122.8852	126.3362	133.9693
6	100.5010	120.1336	129.0031	131.0350	135.1029	142.0987
7	107.8701	128.9713	138.7053	140.3192	144.9082	150.3535
8	114.3327	137.0794	145.5136	147.8620	151.7646	159.0364
9	121.7272	144.6708	154.7982	157.4919	160.7810	168.6182
10	128.7349	152.9570	163.0010	166.2905	169.3529	176.4748
11	136.9179	161.5221	172.1821	175.3291	178.7967	186.0942
12	143.0285	169.3807	180.2560	182.1540	186.8779	194.2016
13	149.5902	177.9830	187.9096	191.0126	195.5303	202.0678
14	158.0408	186.6083	198.2371	201.7923	205.0219	212.4405
15	165.2590	194.2375	206.5662	209.2172	213.1823	219.8407
16	173.0454	203.0749	215.0816	217.4856	220.3072	227.5742
17	179.3371	210.5368	222.3453	225.6316	229.3309	235.3441
18	188.2014	220.5495	231.9111	233.6932	238.3172	245.1766
19	194.0988	227.8202	239.8604	242.7204	246.4353	253.1864
20	203.6118	237.2023	249.2740	251.8718	255.4112	262.5605
21	211.7907	246.0937	258.6246	261.3748	264.0440	270.4297
22	219.8975	255.2739	267.3053	269.8192	273.3937	280.7072
23	226.8004	263.7501	275.4572	278.6257	281.6886	289.8051
24	236.0165	272.0377	284.8483	288.0743	289.9856	297.9279
25	243.0889	282.5232	295.4275	296.4088	300.8389	307.0539
26	253.7127	292.7961	304.4774	307.5744	310.0954	315.6405
27	261.1343	299.7650	311.2469	314.6049	317.6979	322.6385
28	269.4547	309.1641	322.4268	324.7590	327.2547	332.9363
29	277.9067	317.9956	329.7712	332.4295	334.8528	340.2266
30	287.1657	328.5346	341.5482	343.8064	345.3277	350.5673
31	297.5729	337.9642	350.3327	352.3903	353.8502	359.0556
32	305.3029	345.7488	359.8491	360.2178	363.1237	367.4391
33	314.8282	356.6980	369.6890	369.8393	372.4225	376.7413
34	324.4652	366.3151	378.4167	378.2029	380.8598	385.7413
35	334.4574	375.3354	388.8010	390.9910	391.2362	393.9228
36	343.5272	383.6141	397.1437	397.9214	399.2224	401.6898
37	351.7478	393.1965	406.7028	407.1157	407.5269	411.6026
38	362.5028	403.1014	416.3320	418.3173	418.5770	421.3554
39	373.2334	413.8495	427.0818	428.5969	428.1623	432.5115
40	382.9450	424.6684	437.2881	438.6393	438.8290	440.5673
41	392.6071	433.4896	447.0351	447.6415	448.4028	450.5998
42	402.8465	441.9363	456.4540	456.7807	457.7534	459.2062
43	412.7161	452.6886	467.2235	467.2280	467.0481	469.6938
44	422.7184	460.8041	477.1734	476.2853	476.6146	477.7239
45	433.5034	472.0808	485.6310	486.0177	485.4702	487.5539
46	442.3113	481.4498	494.7578	496.0935	495.9943	498.6996
47	454.4903	491.1692	505.0525	505.4172	505.1486	507.9029
48	462.6673	500.6318	516.0816	514.7325	515.4032	517.5349
49	472.1505	509.1696	523.0532	523.6033	523.9010	528.0483
50	482.9520	520.4733	533.1283	533.3581	534.1450	537.9038

Table 6. Table S6: Numerical results from the left plot of Fig. 3 (b) in the main text.

	F1-score: low-field regime
Nuclear spins	8 hours	8 hours	6.4 hours	4.8 hours	3.2 hours
1	0.8374	0.6058	0.5973	0.6034	0.6214
2	0.8461	0.8439	0.8437	0.8456	0.8437
3	0.8523	0.9043	0.9008	0.9001	0.8887
4	0.8354	0.9085	0.9007	0.8973	0.8775
5	0.8035	0.8914	0.8786	0.8741	0.8474
6	0.7648	0.8669	0.8499	0.8426	0.8068
7	0.7249	0.8427	0.8200	0.8068	0.7648
8	0.6855	0.8178	0.7869	0.7705	0.7193
9	0.6434	0.7918	0.7529	0.7330	0.6745
10	0.6064	0.7648	0.7209	0.6958	0.6313
11	0.5674	0.7380	0.6865	0.6599	0.5878
12	0.5325	0.7136	0.6573	0.6250	0.5503
13	0.4993	0.6901	0.6279	0.5934	0.5163
14	0.4722	0.6679	0.6007	0.5622	0.4820
15	0.4470	0.6473	0.5752	0.5341	0.4534
16	0.4246	0.6272	0.5507	0.5102	0.4271
17	0.4036	0.6079	0.5288	0.4851	0.4031
18	0.3821	0.5898	0.5074	0.4617	0.3826
19	0.3648	0.5706	0.4864	0.4425	0.3626
20	0.3483	0.5539	0.4679	0.4237	0.3442
21	0.3311	0.5380	0.4507	0.4071	0.3273
22	0.3163	0.5226	0.4349	0.3901	0.3137
23	0.3034	0.5074	0.4195	0.3755	0.2991
24	0.2896	0.4932	0.4054	0.3607	0.2882
25	0.2754	0.4782	0.3915	0.3488	0.2759
26	0.2643	0.4654	0.3797	0.3372	0.2663
27	0.2528	0.4523	0.3676	0.3243	0.2557
28	0.2420	0.4411	0.3570	0.3153	0.2478
29	0.2313	0.4273	0.3469	0.3050	0.2384
30	0.2213	0.4151	0.3373	0.2974	0.2327
31	0.2134	0.4033	0.3273	0.2880	0.2252
32	0.2035	0.3928	0.3185	0.2816	0.2200
33	0.1955	0.3826	0.3104	0.2735	0.2135
34	0.1886	0.3717	0.3027	0.2656	0.2070
35	0.1818	0.3620	0.2950	0.2591	0.2030
36	0.1736	0.3522	0.2880	0.2529	0.1980
37	0.1677	0.3441	0.2821	0.2487	0.1939
38	0.1628	0.3349	0.2757	0.2425	0.1899
39	0.1555	0.3259	0.2692	0.2366	0.1860
40	0.1503	0.3181	0.2642	0.2325	0.1819
41	0.1459	0.3104	0.2585	0.2277	0.1786
42	0.1402	0.3015	0.2526	0.2231	0.1758
43	0.1356	0.2938	0.2482	0.2192	0.1734
44	0.1320	0.2868	0.2439	0.2154	0.1697
45	0.1269	0.2792	0.2388	0.2119	0.1656
46	0.1223	0.2726	0.2351	0.2082	0.1639
47	0.1193	0.2656	0.2306	0.2051	0.1632
48	0.1154	0.2601	0.2275	0.2022	0.1600
49	0.1116	0.2524	0.2240	0.1989	0.1585
50	0.1090	0.2460	0.2194	0.1959	0.1564

Table 7. Table S7: Numerical results from the middle plot of Fig. 3 (b) in the main text.

	MAE of $A^z$ (Hz): low-field regime
Nuclear spins	8 hours	8 hours	6.4 hours	4.8 hours	3.2 hours
1	260.2282	405.1872	396.5828	390.8325	384.3432
2	251.1654	182.7176	185.0691	187.3382	190.0951
3	263.6726	144.0318	148.6022	149.6183	154.8042
4	293.0560	147.2674	155.4596	157.1728	160.7815
5	329.6712	161.3518	169.5946	173.4622	174.8430
6	368.8992	176.8328	190.5612	190.6899	193.2514
7	404.1857	191.8234	205.8041	205.4915	209.0273
8	443.4137	208.1071	226.6060	227.1567	226.2190
9	478.4960	219.8994	245.3275	242.6228	245.2669
10	518.7376	232.6298	263.0389	261.2364	264.8836
11	549.4742	246.0282	281.8052	280.9841	285.0225
12	581.7095	258.6767	298.6365	296.5463	303.6625
13	608.0917	271.9054	316.6907	318.8004	328.7630
14	631.0861	281.2283	333.3029	335.7496	347.7719
15	648.6327	296.6566	352.3886	357.3402	368.8635
16	673.4216	308.8433	373.2958	381.8805	396.7316
17	681.8454	320.9221	388.0544	399.0695	418.3625
18	700.6658	335.8895	411.7712	422.2548	442.3441
19	712.3122	347.8367	427.5565	440.6676	468.1088
20	721.7626	360.6582	450.9222	462.0290	489.9581
21	734.6053	375.8439	467.2184	483.7496	516.0489
22	735.8680	385.7603	489.0927	507.5904	539.5339
23	747.8732	401.8079	511.5128	531.4620	567.1509
24	756.6541	415.0595	530.3218	557.9933	597.4008
25	760.4934	423.2896	551.6345	574.0417	620.5144
26	768.5827	438.0793	568.3888	594.6057	637.5392
27	772.6880	450.0879	589.9317	618.7290	671.4230
28	780.4636	462.7080	607.4707	639.1926	693.0527
29	788.0663	472.2636	626.9795	656.2526	712.8001
30	791.1951	487.3765	649.8318	683.6196	734.2033
31	797.9576	494.4483	667.8596	702.6072	759.4413
32	803.4013	509.5233	682.4961	724.4530	783.1126
33	807.7159	520.2050	708.1764	741.4894	808.9540
34	816.3036	531.5855	724.5666	764.0583	826.3106
35	826.1163	543.7899	742.5083	782.4291	849.5395
36	834.3168	554.2415	760.9621	798.1111	872.9266
37	837.6385	565.7272	775.1228	819.9009	887.4358
38	844.5581	574.8729	792.9153	839.5465	909.1770
39	853.1921	587.2261	808.1229	853.7594	933.6187
40	852.5966	597.9441	827.3008	868.9729	946.3203
41	860.9590	603.1139	838.1088	888.7731	964.0512
42	878.5686	617.8083	854.4698	909.1423	988.8189
43	878.9279	629.9923	877.2017	930.5638	1010.2498
44	885.5083	642.6727	899.0703	940.4580	1022.6586
45	894.0113	648.4728	909.9989	953.9354	1035.7232
46	895.7027	658.8026	921.4532	970.7240	1047.3836
47	923.8292	665.8942	934.6619	985.8349	1072.5958
48	923.4695	679.2984	948.7267	1000.3493	1077.2146
49	923.8201	690.8374	962.5757	1013.0872	1086.9868
50	939.2969	696.6918	977.2833	1028.6738	1108.2456

Table 8. Table S8: Numerical results from the right plot of Fig. 3 (b) in the main text.

	MAE of $A^{\perp }$ (Hz): low-field regime
Nuclear spins	8 hours	8 hours	6.4 hours	4.8 hours	3.2 hours
1	273.8330	380.2625	374.4611	367.6944	358.3669
2	283.0027	235.4731	236.8377	236.5080	238.0131
3	297.1525	201.2183	205.7130	205.7466	207.8186
4	317.3713	208.5142	214.7527	213.3380	213.3987
5	340.1957	225.4855	231.0126	229.8937	226.5270
6	367.2399	243.2514	253.1049	247.9331	245.6396
7	389.2218	256.1073	267.2818	260.8808	259.0352
8	413.0394	274.0824	285.5685	280.6923	271.6389
9	431.5511	285.2878	301.4492	293.2219	284.8038
10	458.5880	297.9842	316.8277	306.3010	299.6389
11	472.5385	307.4897	330.9316	323.1101	311.7457
12	493.3198	320.9309	347.0050	336.4758	327.9559
13	509.1891	331.1269	363.3086	354.8031	346.6547
14	521.8324	341.0750	376.3447	368.1503	363.3562
15	539.5297	354.5445	395.5978	383.1231	378.9422
16	553.7653	362.5189	409.0987	401.6849	401.8835
17	564.1639	377.3383	427.4721	418.7478	422.9385
18	576.9198	390.4811	446.8978	439.4152	443.9845
19	591.7262	402.7148	461.4408	458.7058	463.7754
20	595.8889	413.8770	480.8585	475.3068	485.9680
21	602.4835	425.4929	497.3971	500.0390	511.4357
22	608.1923	437.1044	518.4661	519.2059	533.3241
23	620.3685	452.5285	537.8783	540.7091	563.0763
24	627.9462	466.4888	559.1950	562.6841	586.2156
25	635.0877	478.1557	577.8161	581.9690	608.6732
26	641.3634	492.4579	595.1573	604.5061	631.3480
27	649.7685	501.8825	615.7656	625.8019	659.7096
28	651.4775	517.7676	631.5600	646.4787	678.2986
29	657.9097	526.4128	649.9413	665.9168	699.0617
30	658.8850	541.2504	672.7703	691.5914	718.9885
31	666.2096	548.4030	691.0638	708.6297	749.2924
32	675.9387	567.0924	711.4765	733.4782	772.8354
33	679.1565	579.0807	732.7810	751.7591	797.2471
34	687.1541	586.9888	747.5967	773.6075	817.5707
35	698.4539	600.3888	766.6566	789.4697	839.3250
36	692.8538	612.8991	784.0538	814.5438	859.2418
37	703.2556	625.6606	804.2034	833.0321	882.2513
38	710.1276	632.8623	816.4343	850.9183	904.7801
39	719.2829	648.0986	834.2477	865.5570	921.9213
40	714.8808	659.0004	851.8566	885.3300	941.8604
41	724.5876	669.1622	862.1095	903.5308	956.5774
42	740.2988	682.0320	882.7659	921.4523	980.2178
43	733.6596	694.7260	900.8817	935.6983	993.5144
44	743.5898	708.9141	923.3632	955.7609	1016.3658
45	746.4862	714.5421	934.6434	969.7010	1025.3020
46	744.6608	727.1051	949.6115	981.2866	1037.2283
47	765.6124	734.6673	960.2760	1000.1270	1061.9457
48	773.0063	746.6041	973.8950	1014.6178	1069.2484
49	768.5114	757.4871	985.8259	1027.1274	1078.7954
50	785.4864	765.5762	1000.7646	1037.4989	1096.2875

Equations

$$\text{EIG}(\tau )\equiv \mathbb{E}\{S\left[p(\vec{A})\right]-S\left[p(\vec{A}|\pm ,\tau )\right]\}.$$

$$P_{\pm }(\tau ):=\int d\vec{A}p(\vec{A})P_{\pm }(\tau |\vec{A}).$$

$$\text{SIG}(\tau )\equiv \mathbb{E}\{{\text{Var}}_{\vec{A}\sim p(\vec{A})}\left[P_{\pm }(\tau |\vec{A})\right]\}.$$

$$\text{SIG}(\tau )=P_{+}(\tau ){\text{Var}}_{\vec{A}\sim p(\vec{A})}\left[P_{+}(\tau |\vec{A})\right]+\left[1-P_{+}(\tau )\right]{\text{Var}}_{\vec{A}\sim p(\vec{A})}\left[1-P_{+}(\tau |\vec{A})\right]={\text{Var}}_{\vec{A}\sim p(\vec{A})}\left[P_{+}(\tau |\vec{A})\right],$$

$$\text{EIG}(\tau )=h\left(P_{+}(\tau )\right)-{\mathbb{E}}_{\vec{A}\sim p(\vec{A})}\left[h(P_{+}(\tau |\vec{A}))\right],$$

$$h(x)=h(x_0)$$

$$-\frac{1}{2\ln 2}\frac{(x-x_0{)}^2}{x_0(1-x_0)}+o((x-x_0{)}^3).$$

$$\text{EIG}(\tau )\propto \frac{{\text{Var}}_{\vec{A}\sim p(\vec{A})}\left[P_{+}(\tau |\vec{A})\right]}{P_{+}(\tau )\left[1-P_{+}(\tau )\right]}.$$

$$\text{EIG}(\tau )\propto \frac{{\text{Var}}_{\vec{A}\sim p(\vec{A})}\left[P_{+}(\tau |\vec{A})\right]}{P_{+}(\tau )\left[1-P_{+}(\tau )\right]+{\sigma }^2}.$$

$$\text{EIG}(\tau )\propto \frac{1}{{\sigma }^2}{\text{Var}}_{\vec{A}\sim p(\vec{A})}\left[P_{+}(\tau |\vec{A})\right],$$

$$H=\sum _{j=1}^n{\omega }_j{\hat{\omega }}_j\cdot {\vec{I}}_j+\frac{f(t)}{2}{\sigma }_z\sum _{j=1}^n{\vec{A}}_j\cdot {\vec{I}}_j,$$

$$P_{+}=\frac{1}{2}\left(1+\prod _{j=1}^n{M}_j\right),$$

$$M_j=1-m_{j,x}^2\frac{(1-\cos {\alpha }_j)(1-\cos \beta )}{1+\cos {\alpha }_j\cos \beta -m_{j,z}\sin {\alpha }_j\sin \beta }\sin {\frac{N{\varphi }_j}{2}}^2,$$

$$\cos {\varphi }_j=\cos {\alpha }_j\cos \beta -m_{j,z}\sin {\alpha }_j\sin \beta ,$$

$$\mathrm{P}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}},\mathrm{R}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}},F1-\mathrm{score}=2\cdot \frac{\mathrm{P}\cdot \mathrm{R}}{\mathrm{P}+\mathrm{R}}.$$

$$\mathrm{MAE}=\frac{1}{n}\sum _{j=1}^n\left|{\vec{A}}_j^{\mathrm{true}}-{\vec{A}}_j^{\mathrm{pred}}\right|.$$

Full text

I Introduction

Quantum sensing experiments are fundamentally limited by the efficiency with which information can be extracted from a physical system. In many platforms, signal acquisition relies on single-photon detection and requires averaging over a large number of experimental repetitions to overcome shot noise, leading to long measurement times that restrict practical applicability [1, 2, 3]. Finding strategies to reduce acquisition time while preserving accuracy is therefore a central challenge for the broad deployment of quantum sensing technologies [4, 5].

A paradigmatic example of this challenge is the detection and characterization of individual nuclear spins [6, 7, 8, 9], a task of primary importance both for extending magnetic resonance imaging to the nanoscale [10, 11, 12] and for developing quantum technologies, where nuclear spins serve as long-lived auxiliary quantum registers for electron spin qubits [13, 14, 15]. Individual nuclear spins can be detected and coherently controlled via their hyperfine interaction with a single electron spin, for example associated to solid-state defects such as nitrogen-vacancy (NV) centers [16, 17], silicon-vacancy centers [18], tin-vacancy centers in diamond [19], and divacancies in silicon carbide [20].

Traditional approaches for nuclear spin cluster characterization rely on long pulse sequences applied to an electron spin to achieve high spectral resolution, resulting in acquisition times that restrict the broad applicability of these techniques [21, 22, 23, 24, 25, 26, 27, 28, 29, 30], particularly when one aims to screen a large number of systems with high throughput [31]. Efficient measurements that preserve accuracy are therefore essential for extending characterization techniques to larger and more complex quantum systems [32, 33].

A promising avenue to reduce measurement time is to identify optimal data acquisition strategies that prioritize experimental settings expected to deliver the highest information gain [5]. These strategies can be non-adaptive, utilizing known prior information about the physics of the system, or can adaptively select the optimal data points based on the outcomes of previous measurements [34, 35, 36, 37]. In this work, we develop a non-adaptive strategy for the selection of measurement data points and benchmark it on an NV sensor. Our approach is built on our recently developed deep learning framework for nuclear spin identification (SALI [38]), which serves as the underlying architecture for processing the experimental signals.

In contrast to Bayesian approaches [39], pre-trained AI architectures such as SALI are tailored to a specific set of measurements and are therefore not straightforward to integrate with sequential adaptive experimental design. They are, however, advantageous due to their microsecond-scale run time. We show how offline experimental design –with measurement settings optimized in advance rather than adapted during the experiment– can still be highly beneficial, exploiting all available prior information on the target parameters to substantially reduce the resources needed for nuclear spin characterization.

Within the framework of Bayesian experimental design, the most well-motivated and widely used criterion is the expected information gain (EIG), defined as the mutual information between the unknown parameters and the measurement outcomes for a given experimental setting. Here, we introduce an alternative figure of merit, the surrogate (expected) information gain (SIG), which is computationally tractable and favors experimental settings that are more robust to measurement imperfections. We argue that EIG can identify experimental settings as informative even when, in practice, they yield little or no information due to unavoidable experimental errors, whereas SIG correctly identifies such settings as uninformative.

Our proposed estimation algorithm consists of three stages. In the first stage, we use the physical system model and a prior distribution of the target parameters, namely the hyperfine couplings between the nuclear spin and the NV center’s electron spin, to determine the control settings most likely to provide the highest information gain, using the SIG as our figure of merit. In the second stage, we simulate measurements at these selected control settings and use the resulting dataset to train the SALI-architecture-based model, which, in contrast to conventional methods that rely on fitting resonances [40], can operate on arbitrary input data. In the third stage, the SALI-based model is applied to real experimental data collected under the same conditions as the simulated training set.

Our work is organized as follows. In Sec. II, we present our offline experimental design methodology for reducing measurement time, introducing the concept of surrogate information gain as a computationally tractable, variance-based strategy rooted in general statistical principles. In Sec. III, we analyze the relationship between EIG and SIG. In Sec. IV, we test the SALI-based model on experimental data to demonstrate its effectiveness in real scenarios. In Sec. V, our methodology for measurement time reduction presented in Sec. II is validated on experimental data in the high-field regime, where the nuclear Larmor frequency $\omega_{\mathrm{L}}$ is much larger than the hyperfine couplings; we also propose a second time-saving strategy that reduces the number of repetitions used to compute each averaged data point, demonstrating an $85\%$ reduction in the total measurement time. In Sec. VI, we validate our methodology in the low-field regime through simulated data, predicting a 60% reduction in measurement time.

II Experimental design and surrogate information gain

We consider a quantum sensor based on a single electron spin ($S=1$), associated to an NV center in diamond, coupled to $n$ ${}^{13}\mathrm{C}$ nearby nuclear spins via the hyperfine coupling $\vec{A}_{j}=(A_{j}^{z},A_{j}^{\bot})$, where $j$ indexes the $n$ nuclear spins in the cluster. The NV electron spin, optically polarized, is prepared in an equal superposition state $\ket{+}\equiv\left(\ket{m_{s}=0}+\ket{m_{s}=1}\right)/\sqrt{2}$. We then apply a dynamical decoupling sequence $\left(\tau-\pi-\tau\right)$ repeated $\mathrm{N}$ times (CPMG sequence), where “$\pi$” represents a $\pi$-pulse on the electron spin, and $\tau$ the inter-pulse delay, followed by single-shot optical spin readout. By repeating this process $\mathrm{N}_{\mathrm{m}}$ times, one can experimentally measure the survival probability $P_{+}(\tau|\vec{A})$ of the initial state. We indicate the complementary probability of measuring the state $\ket{-}\equiv\left(\ket{m_{s}=0}-\ket{m_{s}=1}\right)/\sqrt{2}$ with $P_{-}(\tau|\vec{A})\equiv 1-P_{+}(\tau|\vec{A})$. Together, these two probabilities define $P_{\pm}(\tau|\vec{A})$. The total measurement time required to collect sufficient data to estimate the experimental signal $P_{+}(\tau|\vec{A})$ can be approximated as $t=2\mathrm{N}\cdot\mathrm{N}_{\mathrm{m}}\sum_{j}\tau_{j}$ for a sequence of $\mathrm{N}$ $\pi$-pulses and inter-pulse times $\tau_{j}$. Note that initialization and readout durations are neglected, as they are much shorter: sequences last milliseconds, while initialization and readout take only microseconds.

Experimental design consists of choosing the controllable parameters in an experiment to maximize the information we gain from performing it. The most common approach to experimental design in the context of Bayesian estimation is rooted in information theory and is based on the measurement of those data points that maximize the expected information gain (EIG) [42, 41]. The EIG is defined as the difference in entropy $S$ in the distribution of the unknown parameters (in our case, the vector of hyperfine parameters $\vec{A}\equiv\vec{A}_{j}$, containing the coupling coefficients for all spins) before and after incorporating the results of the measurements:

[TABLE]

The distribution $p(\vec{A})$ is the prior on the parameters, while $p(\vec{A}|\pm,\tau)$ is the posterior, i.e. the distribution of $\vec{A}$ conditioned to the observation of a certain outcome $\pm$ for the chosen inter-pulse delay $\tau$. The expectation value is taken over the possible outcomes $\pm$, computed according to our current knowledge of the system, i.e. the prior $p(\vec{A})$. The outcome probabilities entering the expectation value are given by

[TABLE]

The expression in Eq. (1) corresponds to the mutual information between the parameter vector $\vec{A}$ and the measurement outcome $\pm$, conditioned on the experimental configuration $\tau$. In principle, evaluating the expected information gain directly requires computing the posterior $p(\vec{A}|\pm,\tau)$, which is a computationally expensive task when the dimensionality of $\vec{A}$ is large. However, for a small number of discrete measurement outcomes, it is possible to introduce efficient Monte Carlo expressions that approximate the EIG without requiring evaluation of the posterior. More generally, modern variational techniques make it easier to amortize the cost of evaluating the information gain using neural networks [43, 42]. However, even with these precautions, computing EIG$(\tau)$ for a complex experiment such as the one discussed here can be challenging. We therefore explore alternative approaches that avoid Bayesian or variational methods and instead rely on physically motivated approximations. A further complication in our application is that the number of spins in the cluster is not known a priori, and therefore neither is the dimensionality of the vector $\vec{A}$ (see also [39, 31]).

Under these assumptions, we propose to build an information gain figure of merit from quantities we can easily access, i.e. the model for the signal and the prior on $\vec{A}$, which we refer to as surrogate information gain (SIG):

[TABLE]

In the above expression, the expectation value over the outcomes $\pm$ is computed using the Bernoulli probability distribution in Eq. (2), and the variance is computed over an ensemble of hyperfine parameters distributed like the prior. In other words, for a given measurement configuration $\tau$, we evaluate the variance of the probability of observing certain outcomes, and then compute its expectation value over the possible outcomes. Both the variance over $\vec{A}$ and the expectation value over the results of the measurements can be approximated with a Monte Carlo simulation. In the following, we simplify Eq. (3) by exploiting the fact that the measurement outcome is binary. We explicitly write the expectation value over the measurement outcome:

[TABLE]

from which we see that the expectation value simplifies and the SIG is just the variance of the signal we measure, since the variance of $1-P_{+}(\tau|\vec{A})$ is the same as the variance of $P_{+}(\tau|\vec{A})$. We stress that, although the measurement outcome is binary in the present application, the definition of SIG extends naturally to measurements with an arbitrary number of outcomes through a suitable generalization of Eq. (2). More generally, SIG can be applied to any estimation framework that includes a control parameter, a measurement outcome, and a prior distribution.

We now briefly comment on the interpretation of the SIG. Intuitively, experiments with the highest SIG are those where the signal exhibits the largest variance across the parameter values $\vec{A}$ that are consistent with our prior knowledge. In such cases, the measurement outcomes are most sensitive to differences in $\vec{A}$, making the experiment more informative. Conversely, if for a given inter-pulse delay $\tau$ the variance of the signal is small, then this control adds little value: all plausible values of $\vec{A}$ produce nearly identical signals, so the experiment cannot further refine our estimate. Finally, we observe that, due to the use of $P_{+}(\tau|\vec{A})$ instead of $\log P_{+}(\tau|\vec{A})$ and the non-linearity of the variance, the SIG is not an extensive quantity –that is, it does not increase proportionally as more measurement data are added.

III Relationship between EIG and SIG

In this section, we discuss more formally the relationship between EIG and SIG, and motivate the claim that SIG enables more robust selection of experiments. Assuming that the experiment has a finite number of possible outcomes, which makes it more tractable computationally, one can construct a Monte Carlo estimator for the EIG, which can be further improved through Rao-Blackwellization [42, 44, 45]. For a two-outcome experiment, the Monte Carlo expression for the EIG becomes

[TABLE]

where $h(x)\equiv-x\log_{2}x-(1-x)\log_{2}(1-x)$ is the binary entropy function, $P_{+}(\tau|\vec{A})$ is the outcome probability for a certain experiment, and $P_{+}(\tau)$ is the outcome probability integrated over the prior, defined in Eq. (2). Intuitively, the expression in Eq. 5 is the difference between the binary entropy of a Bernoulli variable with probability $P_{\pm}(\tau)$, and the average over the prior of the entropy of a Bernoulli variable with probability given by the survival probability $P_{+}(\tau|\vec{A})$. We now assume that the fluctuations of $P_{+}(\tau|\vec{A})$ over the prior distribution $p(\vec{A})$ are small, allowing us to perform a Taylor expansion of the binary entropy function as

[TABLE]

This expression is valid for $x_{0}\neq 0,1$, where the binary entropy function is non-differentiable. Substituting it into Eq. (5), we derive the following approximate equation for the EIG:

[TABLE]

The right-hand side of Eq. (7) is bounded by one because of the relation between the variance and the mean of a random variable in $[0,1]$, like $P_{+}(\tau|\vec{A})$. This expression can be interpreted as a signal-to-noise ratio, where signal refers to the fluctuation of the signal over the prior, and noise refers to the shot noise associated to the measurement. In this sense, Eq. (7) states that the information gain for a given measurement configuration is determined by the variance of the signal, rescaled to the shot noise level. This interpretation suggests that a large EIG can be achieved in a regime close to the extrema of the probability interval, i.e. $P_{\pm}(\tau)\simeq 0,1$, where the shot noise becomes very small, even if the variance of the signal is also small. However, such a regime is generally unrealistic, since both the signal and the shot noise are likely to be dominated by background additive noise from the experimental apparatus. The restriction to discrete measurement outcomes, which allows the use of an efficient Rao-Blackwell estimator, also makes it difficult to introduce a more sophisticated noise model based on the addition of Gaussian fluctuations to the measurement outcomes. Intuitively, one would incorporate the strength of the additive Gaussian noise $\sigma$ as follows:

[TABLE]

In the limit in which the noise floor is much larger than the shot noise, we obtain

[TABLE]

which is proportional to the SIG for the binomial outcome. This derivation motivates the definition in Eq. (3) and shows that SIG is expected to be more robust to noise than EIG in its simplest form for finite outcome systems. Although this derivation is carried out for the binomial outcome case, it can be extended in a similar way to multiple outcomes, providing justification for the general definition of SIG.

IV Benchmarking nuclear spin estimation on the full experimental dataset

In this section, we first benchmark the performance of the SALI-based model on the full experimental dataset [28, 30] in the high-field regime (specifically at $B_{z}=404\ \mathrm{G}$). The high-field regime, where the Larmor frequency of the nuclear spins is larger than their hyperfine couplings to the electron spin, is the operating condition of most experiments. Following this validation, in Secs. V and VI, we demonstrate our approach for measurement time reduction.

IV.1 Model training

To train the model, we generate a synthetic dataset by numerically simulating distinct $P_{+}\equiv P_{+}(\tau,\vec{A})$ signals (see the explicit expression for $P_{+}$ in Sec. S1 of the Supplemental Material (SM) [46]). All models in this work have been trained on a dataset of 5 million synthetic samples. These samples were generated based on a prior distribution of the hyperfine coupling constants, where the number of nuclear spins $n$ surrounding the NV detector was uniformly sampled between 1 and 50, and each spin was assigned coupling components $A^{z}$ and $A^{\bot}$ drawn from uniform ranges $A^{z}\in\penalty 10000\ [-50,\ 50]\ \mathrm{kHz}$ and $A^{\bot}\in\penalty 10000\ [2,\ 80]\ \mathrm{kHz}$, respectively.

To account for the effect on the input signals ($P_{+}$) of a weakly coupled spin-bath, comprising thousands of spins with couplings $A^{\bot}\in\penalty 10000\ [0,\ 2]\ \mathrm{kHz}$, we simulated a set of random crystal configurations and computed their respective spin-bath contributions. For each generated sample, one crystal configuration was randomly selected, and its spin-bath contribution was added to the $P_{+}$ signal. For details on the training process, refer to Sec. S2 of the SM [46].

To accommodate the model to the available experimental data, we use as input two $P_{+}$ signals corresponding to pulse sequences with $\mathrm{N}=32$ $\pi$-pulses and inter-pulse spacing $\tau_{32}$ in the interval $[6,\ 50]\ \mathrm{\mu s}$, and $\mathrm{N}=256$ $\pi$-pulses over $\tau_{256}\in[10,\ 40]\ \mathrm{\mu s}$, with temporal resolutions of $\Delta\tau_{32}=\Delta\tau_{256}=4\ \mathrm{ns}$. In addition, NV decoherence effects are modeled as $P_{+}=(1+\mathrm{M}\cdot e^{-\tau/\mathrm{T}})/2$, where $\mathrm{M}$ represents the total contribution from all NV-nuclei couplings [30]. The coherence time $\mathrm{T}$ is computed as $\mathrm{T}^{\mathrm{N}}=\mathrm{T}^{\mathrm{N}=4}\cdot(\mathrm{N}/4)^{\eta}$, with $\mathrm{T}^{\mathrm{N}=4}=3\ \mathrm{ms}$ and $\eta=0.8$. This equation is fitted to experimental data in Ref. [28], characterizing the dependence of coherence time on the number of pulses $\mathrm{N}$. Furthermore, shot noise is simulated by averaging each data point in $P_{+}$ over $\mathrm{N}_{\mathrm{m}}=250$ repetitions of the dynamical decoupling sequences, leading to a cumulative measurement duration of roughly 8 hours. The intervals for $\tau$, temporal resolutions $\Delta\tau$, and number of measurements per data point $\mathrm{N}_{\mathrm{m}}$ reported in this section are taken as the default hyperparameters of the model. Variations of these parameters are explored throughout the paper, as specified in each section.

The ground-truth outputs of the model are 2-dimensional images of size $204\times 160$, where nuclear spins are represented by Gaussian peaks with unit variance, each peak’s position corresponding to the spin’s coupling constants: $A^{z}$ on the y-axis and $A^{\bot}$ on the x-axis.

IV.2 Testing the model with experimental data

We deploy the trained model on the experimental data, obtaining the image on the left of Fig. 1 (a). A total of 29 nuclear spins are identified, 27 of which align with estimations using other techniques [30]. These 27 spins are shown in green boxes in the same image. The remaining two spins, highlighted in red boxes, may correspond to a strongly coupled nuclear spin predicted in Ref. [30], which lies outside our sampling range (beyond the image boundaries). Details of the estimated nuclear spins are reported in Tables S1 and S2 of Sec. S3 in the SM [46].

Leveraging the flexibility of the SALI architecture in handling a varying number of inputs (i.e., a different number of $P_{+}$ input signals), we evaluate its performance when dealing with four inputs. These include the previously used $P_{+}$ data strings, corresponding to sequences with $\mathrm{N}=32$ and $\mathrm{N}=256$ $\pi$-pulses, along with two additional experimental signals that emerge after driving the system with CPMG sequences containing $\mathrm{N}=96$ and $\mathrm{N}=128$ $\pi$-pulses, where $\tau_{96}=\tau_{128}\in[10,\ 30]\ \mathrm{\mu s}$ and $\Delta\tau_{96}=\Delta\tau_{128}=4\ \mathrm{ns}$. Note that the model is retrained to accommodate four inputs. The resulting output is the middle image of Fig. 1 (a), and reveals 30 nuclear spins. Of these, 27 match previously detected spins (highlighted in green boxes), while the effect of the strongly coupled nuclear spin is also evident (nuclear spins highlighted in red boxes).

The performance of the models, shown in Fig. 1 (b), is evaluated using two key metrics. The first one is F1-score, which quantifies the accuracy of the number of detected nuclear spins. This is defined as $2(\mathrm{P}\cdot\mathrm{R})/(\mathrm{P}+\mathrm{R})$, where $\mathrm{P}$ is precision and $\mathrm{R}$ is recall. The second is the Mean Absolute Error (MAE) of the predicted coupling constants $A^{z}$ and $A^{\bot}$ for the detected nuclear spins, which assesses the accuracy of parameter estimation. The definitions of the metrics are provided in Sec. S4 of the SM [46].

V Measurement time reduction: high field

In this section, we benchmark the performance of our procedure for identifying the most informative data points in experimental signals, which are then processed using the SALI-based model. Practically, this allows experimentalists to measure only a subset of data points (those selected by our method), significantly reducing the total measurement time.

V.1 SIG approach: high field

We consider here the high-field regime, where the nuclear spin Larmor frequency is larger than the hyperfine couplings to the central electron spin. In this regime, the signal consists of resonance dips whose positions are determined by the parallel component of the hyperfine coupling $A^{z}$, while their depths are related to the perpendicular component $A^{\bot}$. When multiple nuclear spins are coupled to the NV center, the resonance peaks can overlap in the experimental signal, so that different spins may contribute to a broader dip at the same value of $\tau$. These overlaps make interpretation more challenging, but they also highlight regions of the signal that contain richer information.

To identify these highly informative regions, we apply the SIG approach presented in Sec. II. Specifically, we compute the variance of the $P_{+}$ signal at each data point (i.e., at each $\tau$) across a batch of thousands of synthetic samples drawn from the same prior distribution described in Sec. IV.1. We then select the $\mathrm{N}_{\mathrm{p}}$ data points with the highest variance, prioritizing those that differ the most between samples. This ensures that the AI model relies on sample-specific data points, rather than common ones that do not contribute to distinguishing between different samples.

In Fig. 2 (a), the top plot displays the experimental signal acquired using a CPMG sequence with $\mathrm{N}=256$ $\pi$-pulses. The vertical blue fringes indicate the data points selected by the SIG approach. The bottom plot provides a zoomed-in view of the signal in the region $\tau\in[10,\ 15]\ \mathrm{\mu s}$, where vertical dashed lines mark multiples of $\frac{\pi}{2\omega_{L}}$, with $\omega_{L}$ being the Larmor frequency of the nuclear spins. Around odd multiples of $\frac{\pi}{2\omega_{L}}$, one can observe resonances from the weakly coupled spin-bath ($A^{\bot}\in[0,\ 2]\ \mathrm{kHz}$), which lies outside our target sampling range used in the computation of the SIG. Consequently, the data points near the Larmor frequencies do not contribute to our characterization task. This highlights that the selected data points effectively emphasize the most informative regions for identifying nuclear spins within the desired coupling ranges, $A^{z}\in[-50,\ 50]\ \mathrm{kHz}$ and $A^{\bot}\in[2,\ 80]\ \mathrm{kHz}$, as described earlier. Fig. 2 (b) shows the result of applying this selection method to the full experimental signal. The same approach is applied independently to the signal corresponding to $\mathrm{N}=32$ $\pi$-pulses. In Fig. 2 (c), the model is trained using a synthetic dataset generated by simulating $P_{+}$ signals only at the data points selected by the SIG approach. The resulting model –referred to as the operational model– is then deployed to characterize nuclear spin systems from experimental signals.

V.2 Experimental validation of the SIG approach

We apply the SIG approach to data points within the ranges $\tau_{32}\in[6,\ 50]\ \mathrm{\mu s}$ and $\tau_{256}\in[10,\ 40]\ \mathrm{\mu s}$, with time resolutions $\Delta\tau_{32}=\Delta\tau_{256}=4\ \mathrm{ns}$. The variance is computed across 50,000 synthetic samples, and the $\mathrm{N}_{\mathrm{p}}=4000$ data points with the highest variance are selected from each signal ($\mathrm{N}=32$ and $\mathrm{N}=256$). This selection leads to a reduction in measurement time of approximately $50\ \%$. The model is trained using these selected data points and subsequently tested on the experimental data. The resulting output image is the right image of Fig. 1 (a). The model predicts 27 nuclear spins, of which 25 match previous predictions (green boxes). This suggests that, although the model’s performance declines slightly, as shown in all plots of Fig. 1 (b), the reduction in the number of detected nuclear spins is minimal –only two fewer than in the full-data case. For many characterization tasks, including those needed to construct a quantum memory, this number remains sufficient [48, 32, 47].

Notably, this result is achieved with half the measurement time compared to the full dataset, yet with only a minimal loss in performance: the model detects 25 nuclear spins instead of 27. This demonstrates that the selected data points retain the most informative features of the signal and are sufficient for accurate system characterization.

V.3 Robustness to increased shot noise

Another strategy for measurement time reduction is increasing the shot noise in the input signals. In particular, we analyze the cases with $\mathrm{N}_{\mathrm{m}}=200$, $\mathrm{N}_{\mathrm{m}}=150$, and $\mathrm{N}_{\mathrm{m}}=100$. We generate the same test set as in the original dataset but modify the number of measurements used to compute the average at each data point, reducing it from $\mathrm{N}_{\mathrm{m}}=250$ to the values specified above. These modified test sets are then evaluated using the model in which the measurement time had already been reduced to 4 hours through the SIG approach. This model was trained using $\mathrm{N}_{\mathrm{m}}=250$ measurements.

In Fig 3 (a), we observe that for $\mathrm{N}_{\mathrm{m}}=200$ (3.2 hours) and $\mathrm{N}_{\mathrm{m}}=150$ (2.4 hours) the model’s performance remains nearly identical to that with $\mathrm{N}_{\mathrm{m}}=250$ (4 hours). In the case of $\mathrm{N}_{\mathrm{m}}=100$ (1.6 hours), the model’s performance slightly decreases. Despite this minor degradation, the accuracy of the system’s characterization remains largely unaffected. This measurement-reduction proposal suggests that precise system characterization is possible with just $\mathrm{N}_{\mathrm{m}}=100$ measurements per data point, resulting in significantly faster data acquisition.

VI Measurement time reduction: low field

In this section, we focus on the low-field regime ($B_{z}=40.4\,\mathrm{G}$), which is one order of magnitude lower than the high-field case. In this regime, the signal becomes significantly more complex, as $P_{+}$ no longer consists of a sequence of dips. In particular, we find that distinct samples containing more than ten nuclear spins can produce very similar signals, making them difficult to distinguish (see Fig. S1 in Sec. S5 of the SM [46]). Despite the increased complexity of the data analysis, this regime is experimentally attractive, as applying and aligning smaller magnetic fields is typically much simpler and more convenient. In this scenario, we evaluate the methodology introduced in Sec. II only on synthetic data generated from simulations.

To assess the effectiveness of the SIG approach at low magnetic field, we first train the SALI-based model using 2 input signals constructed identically to those in the high-field regime (see Sec. IV.1); this serves as our reference model. Acquiring these two input signals experimentally would take approximately 8 hours in the lab, taking into account that each data point is measured $\mathrm{N}_{\mathrm{m}}=250$ times.

All other datasets at low field, apart from the reference, are generated from selected $\tau$ values within the range $\tau\in\penalty 10000\ [1,\ 50]\ \mathrm{\mu s}$, with a resolution of $\Delta\tau=1\ \mathrm{ns}$. Improving the resolution of the measurements serves as a counterbalance to the reduced magnetic field. Based on the SIG approach, we select the $\mathrm{N}_{\mathrm{p}}=8000$ data points with the highest variance within the specified $\tau$ range to construct the input signals for pulse sequences with $\mathrm{N}=32$ and $\mathrm{N}=256$ $\pi$-pulses. The total measurement time an experimentalist would need to acquire these selected data points in the lab is approximately 8 hours, which matches the time required to measure the input signals used in the reference model. The model trained on this higher-resolution data, selected via SIG, outperforms the reference model while requiring the same measurement time.

In all plots of Fig. 3 (b), the yellow line represents the reference model, while the black line corresponds to the model incorporating the SIG-based selection of data points. We test the SIG-incorporating model’s resilience to shot noise using synthetic data. Measuring each selected data point $\mathrm{N}_{\mathrm{m}}=100$ times would amount to a total measurement time of about 3.2 hours. Despite this reduced time, Fig. 3 (b) shows that the performance of the 3.2-hour model is comparable to –or even better than– that of the reference model, which requires 8 hours. Again, we do not train a different model for each $\mathrm{N}_{\mathrm{m}}$.

Although characterization at low magnetic field remains significantly less effective than at high field, this approach serves as a step toward optimizing data collection strategies to enhance model performance in regimes that were previously thought to be impractical –both in terms of magnetic field strength and the required number of repetitions per decoupling sequence. By applying the SIG approach and measuring for 8 hours, the model can detect up to 20 nuclear spins, bringing it closer to practical applications in real characterization problems.

VII Conclusions

In this work, we have proposed and demonstrated an offline strategy to optimize data acquisition for the characterization of nuclear spin clusters with a quantum sensor, an inherently time-consuming task. Our approach is based on a newly introduced figure of merit for information gain: the surrogate information gain (SIG), a computationally tractable, variance-driven quantity that is robust to experimental imperfections and can be used to reduce experimental time while maintaining high informational value. This optimization framework was implemented using a deep learning signal-to-image (SALI) architecture, chosen for its microsecond scale run-time and adaptability to varying input selections; our approach is, however, very general and could be readily extended to other estimation methods, for example, based on Markov Chain Monte Carlo [31]. We validated our strategy experimentally in the high-field regime and demonstrated its effectiveness theoretically using synthetic data in the low-field regime. In both regimes, our approach yielded substantial time savings. In the high-field regime, the measurement time –defined as the total experimental time required to measure the signals used by the SALI-based model to characterize the nuclear spin cluster– was reduced by 85$\%$, from 11 hours to 1.6 hours; in the low-field regime, by 60$\%$, from 8 hours to 3.2 hours. These results demonstrate that statistics-guided measurement selection can significantly improve the efficiency of nuclear spin characterization.

Our proposed strategy is non-adaptive, i.e. it does not require any real-time adaptation based on previous measurement outcomes. This makes it intrinsically easy to implement, as it does not require complicated real-time electronics. An open question for future studies is whether the data acquisition time can be further minimized by deploying real-time adaptive techniques [5, 34, 35]. Our parallel work from Ref. [39] shows that variational Bayesian inference, while featuring a slower run-time than the SALI-based model discussed here, can compute a posterior probability distribution within minutes and can be applied to input measurements that are not pre-defined but instead selected adaptively in real time. This is very promising for the implementation of real-time adaptive techniques, and future work will investigate if the SIG, or other figures of merit or computational approaches [42] will enable further reduction of the data acquisition and processing time.

Our results impact quantum technologies in several ways. First, they enable fast characterization of the local environment that leads to decoherence of electron spin qubits. The identification of the nuclear spins in the environment may enable the design of tailored control sequences to minimize decoherence and improve gate fidelities [49]. Second, nuclear spins are a valuable resource for quantum information processing. Owing to their long coherence times, they have been proposed, and in some cases demonstrated, as auxiliary qubits in quantum networking and computing architectures [13, 20, 14, 15, 50]. The design of nuclear spin control and readout protocols requires precise knowledge of their hyperfine coupling to the central electron spin. Our techniques offer a path toward scaling these technologies to larger qubit arrays. Finally, our work may have a significant impact on the field of nanoscale nuclear magnetic resonance (nano-NMR). Reducing the sensing time can make nano-NMR more practical and broadly applicable [10, 11, 12, 51, 52, 53]. In particular, for unstable chemical or biological samples, only a shorter data acquisition time might make the measurements possible.

Although our focus here has been on the identification of individual nuclear spins with a quantum sensor, the proposed SIG approach is generally valid and could be applied to different experiments in multi-parameter quantum systems. For instance, SIG could be employed in multiphase estimation on a photonic circuit, where the controls are tunable phases in the interferometer [55, 54]. Another potential application is waveform estimation in quantum sensors, which could benefit from SIG in finding optimal controls [56, 57].

VIII Data availability

The experimental data analyzed during this study was provided by M. Abobeih and T. H. Taminiau. The synthetic datasets generated for training the neural network are available from the corresponding author upon reasonable request.

IX Code availability

The code used in this study is available from the corresponding author upon reasonable request.

X Acknowledgements

The authors acknowledge discussions regarding numerical strategies with Oliver T. Whaites and Carlos Munuera-Javaloy. C. Bonato, E. Garrote, and J. Casanova equally led this work.

XI Author contributions

Numerical analyses were carried out by B. Varona-Uriarte and F. Belliardo. Experimental data was provided by M. H. Abobeih and T. H. Taminiau. C. Bonato, E. Garrote, and J. Casanova supervised the work. B. Varona-Uriarte wrote the manuscript with input from F. Belliardo, C. Bonato, E. Garrote, and J. Casanova. All authors reviewed the manuscript and actively revised sections to improve clarity and presentation.

XII Funding

J. Casanova acknowledges the Agencia Estatal de Investigación via the Modelizado, Optimización, y Esquemas de Magnetometria en Centros de Color project PID2024-161371NB-C22, and the Basque Government under Grant No. IT1470-22. J. Casanova, E. Garrote and B. Varona-Uriarte acknowledge the Quench project that has received funding from the European Union’s Horizon Europe - The EU Research and Innovation Programme under grant agreement No 101135742. C. Bonato acknowledges support by the Engineering and Physical Sciences Research Council (EP/S000550/1, EP/V053779/1, EP/Z533208/1, EP/Z533191/1, UKRI2706) and the European Innovation Council (QuSPARC, grant agreement 101186889). This work is also supported by the project 23NRM04 NoQTeS, which has received funding from the European Partnership on Metrology, co-financed from the European Unions Horizon Europe Research and Innovation Programme and by the Participating States.

XIII Competing interests

The authors declare no competing interests.

XIV Additional information

Correspondence and requests for materials should be addressed to Borja Varona Uriarte.

**Supplemental Material:

Reducing Sensing Time through Offline Experimental Design for Nuclear Spin Detection**

S1 Description of the system

We consider a system consisting of an NV sensor coupled to $n$ ${}^{13}\mathrm{C}$ nearby nuclear spins through hyperfine interactions characterized by coupling vectors $\vec{A}_{j}=(A_{j}^{z},A_{j}^{\bot})$. The dynamics of the system are described by the Hamiltonian

[TABLE]

where the effective nuclear spin frequency vector is given by $\omega_{j}\ \hat{\omega}_{j}=\gamma_{\rm{n}}B_{z}\ \hat{z}+\frac{1}{2}\vec{A}_{j}$. Here, $\gamma_{\rm{n}}=(2\pi)\times 10.705$ MHz/T is the gyromagnetic ratio of the nuclear spins, and $B_{z}$ is an external magnetic field aligned along the NV axis ($\hat{z}$), defining the Larmor frequency $\omega_{\rm{L}}=\gamma_{\rm{n}}B_{z}$. Microwave (MW) $\pi$-pulses are applied to the NV following the CPMG sequence with inter-pulse spacing $\tau$, and the NV state is readout optically. The modulation function $f(t)=\pm 1$ describes the effect of the applied MW driving. By repeating this sequence $\mathrm{N}_{\mathrm{m}}$ times, the survival probability $P_{+}$ of the NV’s initial state can be computed. In the theoretical limit of infinite measurements (note our simulations consider shot noise and decoherence effects), $P_{+}$ takes the form:

[TABLE]

where

[TABLE]

[TABLE]

with $m_{j,z}=\frac{(A_{j}^{z}+\omega_{L})}{\tilde{\omega}_{j}}$, $m_{j,x}=\frac{A_{j}^{\bot}}{\tilde{\omega}_{j}}$, $\tilde{\omega}_{j}=\sqrt{(A_{j}^{z}+\omega_{L})^{2}+{A_{j}^{\bot}}^{2}}$, $\alpha_{j}=\tilde{\omega}_{j}\tau$, $\beta=\omega_{L}\tau$. See the full derivation in Ref. [1].

S2 Training of the models

The SALI architecture is detailed in the Supplemental Material of Letter [2], with the following modifications. The number of filters in the 1-dimensional and 2-dimensional convolutional layers has been increased from 16 and 32 to 32 and 64, respectively. Additionally, the input data normalization has been replaced by a batch normalization layer at the network’s input.

All models in this study were trained on 5 million samples, divided into 70 % for training, 15 % for validation, and 15 % for testing. Separate models were trained for each data point selection: the 2-input model, the 4-input model, and the SIG-incorporating model at high magnetic field, as well as the reference model and the SIG-incorporating model at low magnetic field. The network was trained using the Mean Square Error (MSE) loss function, calculated across all pixels in the output grid, and optimization was performed with the Adam optimizer, initialized with a learning rate of $1\times 10^{-3}$. The learning rate was reduced during training, with a patience of 2 and a learning rate factor of 0.1. The neural network was implemented in PyTorch and trained for approximately two days on a single NVIDIA H100 80GB GPU.

S3 Nuclear spin predictions

To evaluate the reliability of the predicted nuclear spins, a small Gaussian noise (mean = 0, standard deviation = 0.005) was added to the experimental signals. This simulates classical noise sources present in real measurements, beyond the shot noise already accounted for in the simulations of the $P_{+}$ signals. Each model configuration (2 inputs, 4 inputs, and SIG) was tested 10 times, with different random noise added in each run. The mean values and standard deviations of the predicted coupling constants –corresponding to the images shown in Fig. 1 (a) of the main text– were computed across these 10 runs and are reported in Tables S1 and S2.

S4 Definitions of the metrics

To evaluate the performance of the deep learning model in detecting the number of nuclear spins, we use standard classification metrics: precision (P), recall (R), and the F1-score. These metrics are based on three types of outcomes when comparing predicted spins to the ground truth. A true positive (TP) is a predicted spin that correctly matches a ground-truth spin. A false positive (FP) is a predicted spin that does not correspond to any ground-truth spin. A false negative (FN) is a ground-truth spin that the model failed to detect. Precision measures the fraction of predicted spins that are correct, while recall measures the fraction of ground-truth spins that are successfully detected. The F1-score is the harmonic mean of precision and recall, providing a single metric that balances both false positives and false negatives. The mathematical definitions of the three metrics are:

[TABLE]

In addition to evaluating the detection of the number of nuclear spins, we also assess the accuracy of parameter estimation by computing the Mean Absolute Error (MAE) of the predicted coupling constants. This metric quantifies the average absolute deviation between the predicted values $\vec{A}_{j}^{\,\,\mathrm{pred}}$ and the corresponding ground-truth values $\vec{A}_{j}^{\,\,\mathrm{true}}$, computed across all correctly identified spins (true positives):

[TABLE]

All metrics are computed across all nuclear spins within the test samples, and the plots in Fig. 1 (b) and Fig. 3 in the main text show the average values for samples containing from 1 to 50 nuclear spins.

S5 Low-field regime signals

S6 Numerical results from the main text: metrics