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Comparing Experiments to the Fault-Tolerance Threshold

Richard Kueng, David M. Long, Andrew C. Doherty, Steven T. Flammia

arXiv:1510.05653v2quant-ph

TL;DR

The paper addresses the difficulty of comparing experimentally measured average error rates with worst-case quantities used in fault-tolerance thresholds. It derives relations between these measures for physical noise sources and finds that coherent errors can produce the largest mismatch, while unitarity-based criteria identify when linear scaling is possible.

Abstract

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Achieving error rates that meet or exceed the fault-tolerance threshold is a central goal for quantum computing experiments, and measuring these error rates using randomized benchmarking is now routine. However, direct comparison between measured error rates and thresholds is complicated by the fact that benchmarking estimates average error rates while thresholds reflect worst-case behavior when a gate is used as part of a large computation. These two measures of error can differ by orders of magnitude in the regime of interest. Here we facilitate comparison between the experimentally accessible average error rates and the worst-case quantities that arise in current threshold theorems by deriving relations between the two for a variety of physical noise sources. Our results indicate that it is coherent errors that lead to an enormous mismatch between average and worst case, and we quantify how well these errors must be controlled to ensure fair comparison between average error probabilities and fault-tolerance thresholds.

Quantum states and operations

This section introduces quantum channels and representations used to relate average fidelity to error rates, including a generalized formula for completely positive maps that need not preserve trace.

  • Quantum states are density operators, while quantum operations are completely positive maps acting on them.Source: Quantum states and operations · p. 6
  • Kraus, Liouville, and Choi representations encode the same quantum operation for different analytical purposes.The Choi matrix is positive semidefinite for every completely positive operation, unlike the Kraus and Liouville operators, which need not have complete eigenvector sets.Source: Quantum states and operations · p. 6
  • The average-fidelity formula is generalized to completely positive maps that are not necessarily trace preserving.For trace-preserving maps, the formula reduces to earlier results [30], while the generalization supports leakage-process fidelity estimates.Source: Quantum states and operations · p. 6
  • Twirling preserves average fidelity and reduces completely positive maps to depolarizing operations, simplifying the proof of the generalized formula.The reduction uses Haar-measure invariance, Schur’s Lemma, and direct evaluation of the depolarizing map’s Liouville representation.Source: Quantum states and operations · pp. 6–7

Semidefinite Programming

This section presents semidefinite programming as a convex optimization framework and explains how primal-dual feasibility provides computable certificates and, under equality, exact optima.

Semidefinite Programming

  • A semidefinite program maximizes a linear function over a convex subset of the positive-semidefinite cone.It is specified by Hermitian matrices and a linear map, with matrix inequalities defining the feasible region.Source: Semidefinite Programming · p. 8
  • Semidefinite programming reduces many optimization problems to forms that can be solved efficiently numerically and sometimes analytically.Source: Semidefinite Programming · p. 8
  • Primal and dual semidefinite programs provide complementary optimization formulations linked by weak duality.Every feasible primal objective is bounded above by every feasible dual objective, while strong duality typically makes the optimum values coincide.Source: Semidefinite Programming · p. 8
  • Feasible primal and dual points yield certificates that bound an optimum, and matching objective values certify an analytically exact solution.Source: Semidefinite Programming · pp. 8–9

Semidefinite programs for the diamond distance

The diamond distance between quantum channels can be characterized by a primal-dual pair of semidefinite programs, enabling bounds or exact values from feasible constructions.

Semidefinite programs for the diamond distance

Dephasing and calibration errors for a single qubit

For a single-qubit dephasing and calibration-error channel, the Kraus representation supports evaluation of average fidelity and matching diamond-distance bounds.

Dephasing and calibration errors for a single qubit

Thermal relaxation of a single qubit

The paper derives a bound for the diamond distance of a one-qubit amplitude-damping channel, improving a prior feasible solution for all parameters p and γ.

Thermal relaxation of a single qubit

  • Theorem 1 gives a diamond-distance relation for the one-qubit amplitude-damping channel valid for every p, γ ∈[0, 1].The bound is obtained by constructing a feasible point of the dual semidefinite program.Source: Thermal relaxation of a single qubit · p. 10
  • The proof computes the Choi matrix of the channel difference and constructs a positive semidefinite dual-feasible operator.The feasibility argument uses constraints on the Choi-matrix difference and an elementary square-root inequality.Source: Thermal relaxation of a single qubit · p. 10
  • The resulting objective value supplies the desired diamond-distance upper bound, which is then established for the full parameter range.The derivation concludes by evaluating the dual objective and verifying the final inequality.Source: Thermal relaxation of a single qubit · pp. 10–11

Incoherent leakage errors

For incoherent leakage, the paper models population transfer into leakage states and derives diamond-distance bounds that depend on leakage probability, projection rank, and the averaging space.

Incoherent leakage errors

  • The model assumes that |1⟩ relaxes to a leakage state |l⟩ with leakage probability p, using a rank-deficient projection P.For d = 2, P corresponds to a pure quantum state and recovers the single-qubit model examined in [34, Eq. (25)].Source: Incoherent leakage errors · pp. 11–12
  • The average error rate depends on whether fidelity is averaged over the qubit subspace or the full three-level space.The full-space expression differs slightly, without qualitative changes; the paper therefore uses qubit-subspace averaging as more physically motivated.Source: Incoherent leakage errors · p. 11
  • The paper exactly determines the diamond distance of the projection-based channel through dual and primal semidefinite-program arguments.A feasible dual point gives the upper bound, while a feasible primal pair gives a matching lower bound.Source: Incoherent leakage errors · p. 12
  • The general bounds vary with projection rank: the upper bound is tight for (d −1)-dimensional projections but becomes looser for more rank-deficient projections.For d = 3 and rank(P) = 2, the resulting relation agrees with the earlier incoherent-leakage model and is slightly stronger.Source: Incoherent leakage errors · p. 12
  • For single qubits (d = 2), the upper bound is tight; more generally, it is saturated for rank-one projectors P.Source: Incoherent leakage errors · p. 12

Coherent leakage errors

The paper analyzes coherent leakage as a unitary coupling between |1⟩ and |l⟩, finding that its diamond distance scales as the square root of the average error rate.

Coherent leakage errors

  • The coherent leakage channel is the unitary U(δ)=exp[−iδ(|1⟩⟨l| + |l⟩⟨1|)], with δ controlling error strength.Projecting onto the qubit subspace yields the average error rate rCL=[2−cos δ−cos^2 δ]/3.Source: Coherent leakage errors · pp. 12–13
  • The channel’s diamond distance is DCL = |sin δ|, and this result is unchanged by whether average fidelity is computed in the projected or three-level space.The equality of the two averaging choices is specific to this coherent leakage model.Source: Coherent leakage errors · p. 13
  • For moderate coherent-leakage errors, the average rate satisfies DCL^2/2 ≤ rCL ≤ 2DCL^2.The upper bound uses δ ∈[−π/2, π/2], while the lower bound follows from 4 sin^2(δ/2) ≥ sin^2 δ.Source: Coherent leakage errors · p. 13

Unitary errors

For d-dimensional unitary channels, the paper derives general bounds relating diamond distance to average error and proves equality for single-qubit unitary channels.

Unitary errors

  • All unitary channels exhibit the unfavorable square-root behavior in which worst-case error is roughly the square root of average error.The paper summarizes this behavior before deriving dimension-dependent bounds.Source: Unitary errors · p. 13
  • Theorem 2 gives dimension-dependent upper and lower bounds on the diamond distance of a d-dimensional unitary channel.The proof uses the channel’s eigenvalue decomposition, a maximally entangled state, and the Choi matrix of the channel difference.Source: Unitary errors · p. 13
  • For single-qubit unitary channels, the lower bound is attained with equality.Global-phase invariance reduces the unitary to a two-dimensional diagonal form, enabling an exact optimization.Source: Unitary errors · pp. 13–14
  • The paper leaves open whether the dimensional dependence in the general upper bound can be improved.The lower bound is tight, but the authors do not establish optimality of the upper-bound dimension dependence.Source: Unitary errors · p. 13
  • The general proof constructs a feasible dual point from the two nonzero eigenvalues of the Choi-matrix difference and uses norm inequalities for the upper bound.For the lower bound, diamond and induced trace norms coincide for differences of unitary channels.Source: Unitary errors · p. 14
  • The same techniques exactly characterize the diamond distance of single-qubit coherent leakage as DCL = |sin(δ)|.This extends the unitary-channel analysis to the three-level coherent-leakage model.Source: Unitary errors · p. 15

The unitarity and average error rate for two-qubit processes

The two-qubit process is analyzed by directly computing its average error rate and unitarity, using simplified formulas because both quantities are analytically accessible.

The unitarity as a witness for unfavorable scaling

The paper uses unitarity to determine when diamond distance scales linearly with average error and when it instead scales as the square root of that error. For unital trace-preserving channels, this criterion can be checked experimentally from r and u.

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