Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States

Nicolas C. Menicucci

Source-linked AI summary

Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States

Nicolas C. Menicucci

arXiv:1310.7596v3Oct 28, 2013quant-ph

TL;DR

The paper models finite-squeezing effects as uncorrelated Gaussian displacements and tracks their evolution through error correction and cluster-state gates. It finds that the controlled-Z gate has the highest error probability among the considered gates, while pure-ancilla envelopes add momentum noise and can produce high error probabilities.

Problem
The paper examines how approximation-induced noise propagates through continuous-variable cluster-state gates and error correction.
Method
It represents physical quadratures as ideal quadratures plus an uncorrelated Gaussian displacement, then evolves the error matrix across single- and two-mode gates.
Results
The controlled-Z gate has the highest error probability among the considered gates, with its quoted value obtained from the error-matrix evolution and correction counts.
Takeaways & Limitations
Using asymmetric initial noise matches sequential gate application, while two-mode error correction prevents correlated errors from building up.

Abstract

from arXiv · show

A long-standing open question about Gaussian continuous-variable cluster states is whether they enable fault-tolerant measurement-based quantum computation. The answer is yes. Initial squeezing in the cluster above a threshold value of 20.5 dB ensures that errors from finite squeezing acting on encoded qubits are below the fault-tolerance threshold of known qubit-based error-correcting codes. By concatenating with one of these codes and using ancilla-based error correction, fault-tolerant measurement-based quantum computation of theoretically indefinite length is possible with finitely squeezed cluster states.

1. GKP error correction using CV cluster states

GKP error-correction circuits can be expressed within the CV cluster-state measurement framework. Ancilla-assisted measurements detect and correct position and momentum shifts while preserving the measurement-based structure.

1. GKP error correction using CV cluster states: GKP error-correction circuits can be rewritten as CV cluster-state computations.The reformulation is equivalent to the original GKP circuits and fits the cluster-state framework.
1. GKP error correction using CV cluster states: Fourier transforms and outcome-dependent displacements connect the cluster implementation to standard GKP correction circuits.These operations are treated as routine components of measurement-based quantum computation.
1. GKP error correction using CV cluster states: Measuring p̂ on three marked nodes leaves a corrected version of the input state at the blank node.The blank node is a p̂-squeezed vacuum state, and each link represents a CẐ gate of weight +1.

2. Evolution of the error matrix

The paper models finite-energy encoded states as ideal GKP states subjected to classical Gaussian displacement noise. Gaussian cluster-state gates propagate this noise linearly through a symplectic transformation, updating its covariance matrix.

2. Evolution of the error matrix: The error matrix η represents the covariance of a random Gaussian displacement applied to an ideal GKP codeword.The displacement is independent of the quantum information encoded in the ideal state.
2. Evolution of the error matrix: The physical quadrature vector is modeled as x̂ = x̂_ideal + y, with y a zero-mean classical random displacement.This separates ideal quantum evolution from statistically characterized displacement noise.
2. Evolution of the error matrix: The Gaussian envelope is irrelevant when every Wigner-function spike has the same error matrix, but becomes relevant for magic-state distillation.The envelope regulates spike heights in the finite-energy encoded state.
2. Evolution of the error matrix: Gaussian unitary evolution maps the quadratures as x̂′ = Sx̂, so the random displacement and ideal state both evolve by the symplectic matrix S.The covariance matrix therefore evolves under the same linear phase-space transformation.

3. Single-mode gates

Single-mode gates are analyzed by propagating an initial error matrix through measurement steps and ancilla-based corrections. After the final correction, the resulting matrix becomes the input noise matrix for the next gate.

3. Single-mode gates: The initial error matrix η0 is propagated through each measured cluster node using a gate-dependent symplectic transformation.Measuring p̂ + m_jq̂ adds cluster-generation noise of variance ϵ in the p̂ quadrature.
3. Single-mode gates: η4,c is the final corrected error matrix and becomes η0 for the next gate, enabling sequential gate analysis.This equality holds for the single-mode gates Î, P̂, and F̂ considered in Table II.
3. Single-mode gates: A single-mode logical error occurs if either correction at steps 3 or 4 fails.The failure probability is computed from the Gaussian shift distribution and its measurement uncertainty.
3. Single-mode gates: Under the assumption δ = ϵ = σ2, the correction-error variances reduce to multiples of σ2 used in the gate calculations.The appendix denotes these multiples by n_jσ2.

4. Two-mode gate (ˆCZ)

The two-mode CẐ gate is analyzed with a 4 × 4 error matrix that tracks inter-mode correlations and additional noise from the implemented gate. Ancilla correction prevents correlated errors from accumulating, while four independent corrections determine the gate-error probability.

4. Two-mode gate (ˆCZ): The CẐ analysis uses a 4 × 4 error matrix because errors may correlate across the two modes.The subsequent p̂ measurements propagate this two-mode covariance matrix through the cluster.
4. Two-mode gate (ˆCZ): Measurements on the vertical cluster segment produce a weight −1 CẐ and add independent p̂ noise of variance ϵ to each mode.The added noise commutes with the CẐ gate, and encoded CẐ is represented by either weight ±1.
4. Two-mode gate (ˆCZ): The implemented CẐ operation first modifies η0 to η′0 by introducing additional noise before teleportation through the cluster.The later measurement evolution therefore starts from η′0 rather than η0.
4. Two-mode gate (ˆCZ): After correction, η0 again equals η4,c and consists of two copies of the single-mode input matrix, preventing correlated errors from building up.The two-mode correction is modeled as two independent corrections at each of steps 3 and 4.
4. Two-mode gate (ˆCZ): Four independent correction instances determine the CẐ error probability, with n3 = 7 and n4 = 5 in the variance calculation.The squares in the success-probability expression arise from two corrections at each step.

1. Method

The method prepares encoded Hadamard eigenstates from an ancilla-supplemented cluster, uses photon counting for destructive identification, and compensates asymmetric noise before measurement.

Encoded Bell pairs are generated by applying the CZ gate to two logical |+L⟩ states through cluster-state measurements.
A random encoded Hadamard on one rail decoheres the Bell pair into the Hadamard eigenbasis with equal probabilities.
Photon counting on one mode identifies the corresponding state on the other using the outcome modulo 4; odd outcomes are impossible ideally.
Because GKP correction produces asymmetric noise, the procedure adds a random Gaussian shift in q before photon counting.
The error probability and success probability are calculated in the Wigner picture under the assumption of pure ancillas.

2. Importance of the Gaussian envelope

The Gaussian envelope regulates finite-energy encoded states and can be represented as a nonunitary cooling operation. It is irrelevant for Clifford error correction but affects photon-counting errors.

The Gaussian envelope regulates spike heights so encoded GKP states have finite energy while retaining a common error matrix.
The envelope is irrelevant to GKP error correction when every spike has the same error matrix, but an overly broad envelope increases photon-counting errors.
A position-space Gaussian envelope corresponds to momentum-space convolution, and the analogous momentum envelope corresponds to position-space convolution.
For large ξ^2, the combined envelopes are represented by a single nonunitary cooling operation up to normalization.
Cooling damps high-number components in the number basis and does not cause photon-counting errors because it is diagonal there.

3. Pure ancilla states

Pure ancillas are modeled through cooling-generated envelopes that avoid photon-counting errors, but the resulting spike variance raises the reported error probability; two modifications could improve it.

Pure ancillas are motivated because cooling preserves purity and introduces no photon-counting error.
The pure-ancilla model sets ξ^2 = 1/2δ, giving blurring variance δ and an envelope variance of 1/4δ.
The correction procedure can shift states toward the origin because minimal codespace shifts leave high-energy spikes far from the center.
Cluster propagation adds blurring without an average Gaussian envelope, leaving states with the ancilla envelope but potentially oversized spikes.
3σ^2 spike variance with δ = ϵ = σ^2 makes the spikes three times broader in variance than the envelope model requires, producing the reported high ε.
The reported probabilities are below the distillation threshold, while recentering corrections or discarding high photon counts could further reduce error probability.

4. Probability formulas

The probability analysis models cooling and photon counting in the Wigner picture, using coarse-grained modulo-4 projectors to compute outcomes, errors, and success rates.

The superoperator M applies an isotropic Gaussian envelope and blurring to model the physical operation used in the analysis.
Photon-counting outcomes are coarse-grained modulo 4 by projectors onto the corresponding photon-number subspaces.
Noise can be moved from the state to the measurement operator, after which outcome probabilities are evaluated for the relevant photon-counting results.
Conditioned on an even photon-counting outcome, the error probability is computed from the probabilities of the competing encoded states.
The success probability is the probability of obtaining an even photon-counting outcome.

5. Wigner-picture probability calculation

The Wigner-picture calculation constructs formal operator representations, regularizes divergent series, and evaluates Gaussian blurring to obtain the quantities needed for the probability calculation.

Operator construction: The calculation begins with four operators indexed by b ∈ Z4 and combines them linearly to construct modulo-4 projectors.The identity corresponds to Φ0, and linear combinations produce the desired projectors.
Regularization: Formal Wigner-function series are introduced for these operators, but some do not converge and therefore require an exponential cooling factor before taking β → 0.The cooled operators provide a regulated route for handling the problematic formal sums.
Regularization: The problematic case b = 2 is handled by redefining its Wigner function through the limit β → 0+.The other cases present no problem under this limiting prescription.
Gaussian blurring: The calculation analytically convolves the relevant Wigner functions with an isotropic Gaussian of variance τ^2 to obtain blurred versions that recover the unblurred expressions as τ^2 → 0.The Wigner trace formula is used to evaluate products of Hermitian operators in this representation.
Projector representation: The desired projector Wigner functions are obtained by combining encoded Pauli and identity Wigner functions into the Hadamard-eigenstate projectors π±.The resulting functions are unnormalized and non-normalizable, with normalization performed within the calculation.