Robust fault tolerance for continuous-variable cluster states with excess anti-squeezing

TL;DR

This paper demonstrates that excess anti-squeezing does not affect the fault-tolerance threshold in continuous-variable cluster states, simplifying experimental requirements for scalable quantum computing.

Contribution

It shows that anti-squeezing has no impact on fault tolerance, removing the need for state purity and guiding experimental squeezing targets.

Findings

Anti-squeezing does not influence fault-tolerance thresholds.

Experimental focus should be on achieving 15-17 dB of squeezing.

Purity requirements can be relaxed in fault-tolerant designs.

Abstract

The immense scalability of continuous-variable cluster states motivates their study as a platform for quantum computing, with fault tolerance possible given sufficient squeezing and appropriately encoded qubits [Menicucci, PRL 112, 120504 (2014)]. Here, we expand the scope of that result by showing that additional anti-squeezing has no effect on the fault-tolerance threshold, removing the purity requirement for experimental continuous-variable cluster-state quantum computing. We emphasize that the appropriate experimental target for fault-tolerant applications is to directly measure 15-17 dB of squeezing in the cluster state rather than the more conservative upper bound of 20.5 dB.

Equations

$$W_{\kappa ,\epsilon }(q,p)$$

$$W_a(q_1,p_1)=W_{\text{in}}(q_1,p_1-m_j{q}_1).$$

$$W_b(\mathrm{x})$$

$${\tilde{W}}_{\text{out}}^{(t)}(q_2,p_2)$$

$$[f\ast {}_{1}g](x,y):=\int dwf(w)g(x-w,y)$$

$$W_{\text{out}}^{\text{avg}}(q_2,p_2)=[G_{\epsilon }\ast {}_1{W}_a](p_2,-q_2).$$

$$W_a(\mathrm{x})$$

$$\times G_{\epsilon }(p_2-q_1-q_3)G_{\epsilon }(p_3-q_4-q_2).$$

$${\tilde{W}}_{\text{out}}^{(r,t)}(q_1,q_4,p_1,p_4)$$

$$=\int dudv{W}_{\text{in}}(q_1,q_4,p_1+u,p_4+v)$$

$$\times G_{\kappa }(t-u)G_{\kappa }(r-v)G_{\epsilon }(v-q_1)G_{\epsilon }(u-q_4).$$

$$W_{\text{out}}^{\text{avg}}(q_1,q_4,p_1,p_4)$$

$$=[G_{\epsilon }\ast {}_4(G_{\epsilon }\ast {}_3{W}_{\text{in}})](q_1,q_4,p_1+q_4,p_4+p_1).$$

Full text

Introduction.—Measurement-based quantum computation (MBQC) employs highly entangled resource states known as cluster states Briegel2001 as a substrate for a quantum computation (QC). A specific computation is carved from the cluster state using only adaptive single-qubit measurements Raussendorf2001 . MBQC eliminates the need for coherent, multi-qubit interactions during the computation, which provides an advantage over circuit-model methods Nielsen2010 .

Continuous-variable (CV) MBQC extends this scheme by utilising CV resources, rather than qubits, to build the initial cluster state Nick2006 . This provides a distinct advantage in scalability over optical-qubit-based MBQC schemes since CV cluster states can be made deterministically on an immense scale Menicucci2008 ; Menicucci2011a . Other MBQC schemes have been able to achieve qubit cluster states of only 6 qubits utilising two photons Ceccarelli2009 , compared with 60 frequency modes Chen:2014jx or $10^{4}$–$10^{6}$ temporal modes for a CV cluster state Yokoyama2013 ; Yoshikawa2016 , albeit with a 1D topology in both cases. Accessible proposals exist for making computationally universal (i.e., 2D) CV cluster states on a similar scale Menicucci2011a ; Wang:2014im ; Alexander2016 ; Alexander2018 .

Still, finite squeezing (required by finite energy) deposits noise that accumulates throughout a computation Alexander2014 ; Gu2009 . Appropriately encoded qubits GKP —available on demand to be coupled into the CV cluster state at will—can survive this noise with regular rounds of quantum error correction. As long as the squeezing—in both the CV cluster state and in the encoded qubits—is high enough, fault-tolerant quantum computation is possible Nick2014 .

The main idea behind that result is to convert the additive Gaussian noise due to finite squeezing Gu2009 to logical-Pauli noise at the encoded-qubit level after every logical-Clifford gate. The quantum-error-correction scheme proposed by Gottesman, Kitaev, and Preskill (GKP) GKP enables this. With the logical-Clifford gates regularly spaced into a grid and supplemented by distillation of magic states Bravyi:2005dx , the problem is effectively mapped to a circuit-model computation using noisy gates Nick2014 . This is a well-studied problem (see Ref. Gottesman:2009ug for a review). The acceptable threshold for gate errors depends on the chosen qubit-level quantum error-correcting code employed to get rid of this residual error.

Figure  1 depicts a CV resource state for universal computation. We describe this as a “flowerbed” comprising two essential parts: (1) a large, canonical Menicucci2011 CV cluster state with a square-lattice graph Gu2009 ; and (2) GKP-encoded $\ket{0_{L}}$ ancillae attached at regular intervals to the cluster-state base by $\hat{\mathrm{C}}_{Z}[1]=e^{i\hat{q}\otimes\hat{q}}$ gates. Performing $\hat{q}$ measurements on unwanted nodes (grey) “carves out” the structure of a desired CV quantum circuit. The remaining nodes are measured in $\hat{p}$ or $\hat{p}+\hat{q}$ (known as a shear measurement Nick2014 ) to enact one- and two- qubit logical Clifford gates. Universality is achieved by distillation of magic states Bravyi:2005dx , which was previously thought to require an additional non-Gaussian resource GKP ; Nick2014 . It was recently shown, however, that distillable GKP magic states can be produced using heterodyne detection—a Gaussian measurement—on a GKP-encoded Bell pair Baragiola2019 .

A single example of both the one- and two-mode gates is shown in yellow and again independently in Fig. 2. The output of one such gate becomes the input of the next. This way, the white nodes in Fig. 1 can be seen as a series of one- or two-mode gates acting in sequence. With this procedure, the one-mode gate is sufficient to enact any single-mode Gaussian unitary gate by a series of shear measurements followed by error correction.

The gate in Fig. 2(b) is required to enact two-mode quantum gates such as the $\hat{\mathrm{C}}_{Z}$ gate. This gate functions by connecting the two input nodes vertically with two nodes to be measured in $\hat{p}$. Measuring these connecting nodes implements a $\hat{\mathrm{C}}_{Z}[-1]=e^{-i\hat{q}\otimes\hat{q}}$ gate on the input states, after which they pass through two one-mode identity gates (which are included only to keep the calculation on a regular lattice Nick2014 ).

One of us Nick2014 has demonstrated that for a pure cluster state (i.e., one created from squeezed vacuum states), there exists a finite squeezing threshold of no higher than 20.5 dB that enables fault-tolerance to be achieved. Here we generalize that result to the case where the CV cluster-state (base of the flowerbed) is built from squeezed thermal states instead of squeezed vacuum states. This introduces additional anti-squeezing, which will turn out—surprisingly—not to affect the threshold calculations at all.

Definitions.—We work with quadrature operators ${\hat{q}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{\dagger})}$ and ${\hat{p}=\frac{-i}{\sqrt{2}}(\hat{a}-\hat{a}^{\dagger})}$ satisfying ${[\hat{q},\hat{p}]=i}$, with ${\hbar=1}$. The vacuum variance is ${\langle{\hat{q}^{2}}\rangle=\langle{\hat{p}^{2}}\rangle=\frac{1}{2}}$. We denote column vectors of position and momentum operators as $\hat{\bm{\mathrm{q}}}$ and $\hat{\bm{\mathrm{p}}}$, respectively, and we collect both into the column vector $\hat{\bm{\mathrm{x}}}\coloneqq(\hat{\bm{\mathrm{q}}}^{\mathrm{T}},\hat{\bm{\mathrm{p}}}^{\mathrm{T}})^{\mathrm{T}}$.

A squeezed vacuum state with squeezing factor ${s>1}$ is a Gaussian state with 0 mean, variance ${{\varepsilon}_{0}\coloneqq\frac{1}{2}s^{-2}}$ along the squeezed quadrature, and variance ${\kappa}_{0}\coloneqq\frac{1}{2}s^{2}$ along the anti-squeezed quadrature. The corresponding squeezing parameter ${r=\frac{1}{2}\ln s}$ so that ${s=e^{2r}}$. Note that a measured variance $\sigma^{2}$ corresponds to $10\log_{10}(2\sigma^{2})$ dB, with negative corresponding to squeezed and positive to anti-squeezed.

A squeezed thermal state is defined here in terms of its measured variances rather than in terms of a squeezing parameter and a temperature. Using this convention, we designate the variance ${\varepsilon}$ along the squeezed quadrature to match the squeezed-vacuum case—i.e., ${\varepsilon}={\varepsilon}_{0}=\frac{1}{2}s^{-2}$, while the variance along the anti-squeezed quadrature is larger than in that case: ${{\kappa}\coloneqq{\frac{1}{2}(s^{2}+\delta^{2})}={\kappa}_{0}+\frac{1}{2}\delta^{2}}$. The additional variance in the anti-squeezed quadrature, $\frac{1}{2}\delta^{2}$, is called the additional anti-squeezing. The Wigner function for this state is

[TABLE]

where $G_{\sigma^{2}}(x)$ is a normalised Gaussian with zero mean and variance $\sigma^{2}$. Note that $\sqrt{{\varepsilon}{\kappa}}\geq\frac{1}{2}$, with equality if the state is pure (squeezed vacuum), and ${\varepsilon}={\kappa}=\frac{1}{2}$ in the case of the vacuum. These states are shown in Fig. 3.

Wigner representation of cluster-state computation.—As quantum information propagates from node to node through a CV cluster state, operations are performed by projective measurements on each node. Due to finite squeezing, noise is introduced at each of these steps, which appears as convolution of one quadrature of the input-state Wigner function when the cluster state is pure (made from squeezed vacuum) Alexander2014 ; Gu2009 .

By tracking how this noise accumulates throughout the computation, we can determine how the additional anti-squeezing might affect the qubit-level error rate. This is achieved, as discussed throughout the supplementary material in Ref. Nick2014 , by evolving the input Wigner function through the appropriate quantum gates according to W(\bm{\mathrm{x}})\xrightarrow{\text{\hat{G}}}W^{\prime}\big{[}\bm{\mathrm{S}}^{-1}_{\hat{G}}(\bm{\mathrm{x}}-\bm{\mathrm{c}})\big{]}, where $\bm{\mathrm{x}}\coloneqq(\bm{\mathrm{q}}^{\mathrm{T}},\bm{\mathrm{p}}^{\mathrm{T}})^{\mathrm{T}}$ is the vector of all phase-space coordinates, and where $\bm{\mathrm{S}}_{\hat{G}}$ and $\bm{\mathrm{c}}$ are found via the Heisenberg action of the Gaussian unitary $\hat{G}$ on the vector of quadrature operators $\hat{\bm{\mathrm{x}}}$—i.e., $\hat{G}^{{\dagger}}\hat{\bm{\mathrm{x}}}\hat{G}=\bm{\mathrm{S}}_{\hat{G}}\hat{\bm{\mathrm{x}}}+\bm{\mathrm{c}}$ Alexander2014 . Quadrature measurements replace the measured variable with its outcome and integrate over the conjugate variable—e.g., measuring $\hat{p}_{1}$ with outcome $s$ maps $W(\bm{\mathrm{x}})\to\tilde{W}_{\text{out}}(\bm{\mathrm{x}}_{\geq 2})\coloneqq\int dq_{1}\,W(\bm{\mathrm{x}})\rvert_{p_{1}=s},$ where $\bm{\mathrm{x}}_{\geq 2}$ is $\bm{\mathrm{x}}$ for the unmeasured modes, and the tilde indicates that the Wigner function is unnormalised.

Results.—We examine four occasions where additional anti-squeezing might affect the cluster-state output: (1) using $\hat{q}$ measurements to delete a node, (2) a one-mode gate, (3) a two-mode gate, and (4) magic state preparation.

(1) Deletion via $\hat{q}$ measurements.—Since each node of the flowerbed is attached to its neighbors by a $\hat{\mathrm{C}}_{Z}[1]$ gate, we only need to use the fact that a $\hat{q}$ measurement after this gate just induces an outcome-dependent momentum shift on the other mode: $({\vphantom{\bra{s}}}_{q_{1}}\!{\bra{s}}\otimes\hat{I})e^{i\hat{q}_{1}\hat{q}_{2}}=e^{is\hat{q}_{2}}=\hat{Z}_{2}(s)$. Since $s$ is known, we can correct it on each neighboring node with $\hat{Z}(-s)$, and the result is the same as if the deleted node had never been attached in the first place. The input state makes no difference to this analysis.