Distance measures to compare real and ideal quantum processes

Motivation

• A standardized error measure is needed to compare experimental processes with ideals and with other implementations, replacing largely ad hoc assessments.Such a measure could also connect experiments to fault-tolerance thresholds.Source: §I · p. 1

Contribution

• The paper addresses the gap between theoretically motivated error measures and experimentally practical measures through a comprehensive survey.Prior work generally treated narrower applications or fewer criteria rather than comparing measures across broad theoretical and experimental concerns.Source: §I · pp. 1–2

Criteria

• Candidate measures should satisfy six properties: metric structure, direct calculation, experimental measurability, physical interpretation, stability, and chaining.Stability makes unrelated ancillary systems irrelevant, while chaining bounds composite-process error by the sum of component errors.Source: §I · pp. 1–2
• Stability and chaining allow component processes to be benchmarked and combined into a conservative bound without full process tomography of a large computation.This is important because tomography of an n-qubit computation requires exponentially many measurements.Source: §I · p. 2

Scope

• Diagnostic measures can reveal sources of experimental imperfections even when they fail the gold-standard criteria, but their detailed investigation is outside this paper’s scope.The paper distinguishes usefulness for diagnosis from suitability as a primary standardized measure.Source: §I · p. 2

Survey strategy

• The survey eliminates many plausible approaches and narrows the promising candidates to J distance, J fidelity, S distance, and S fidelity.Some rejected measures may remain useful diagnostically, but they are unsuitable as primary error measures.Source: §I · pp. 2–3

Recommendations

• Each of the four surviving measures has an operational interpretation for quantum-computation success or failure, while S distance and S fidelity are identified as the most attractive gold-standard candidates.The paper does not choose between those two because their strengths and weaknesses are extremely similar.Source: §I · p. 3

Quantum operations

• Quantum operations encompass unitary evolution, measurement, noise, and decoherence, and map input density operators to output states.Their operator-sum representation uses operation elements that completely describe the process.Source: §II · p. 3
• The paper assumes trace-preserving operations, for which the operation elements satisfy the completeness condition corresponding to physical processes without post-selection.Many results also extend to non-trace-preserving operations.Source: §II · p. 3

Process matrix

• Because the operator-sum representation is not unique, the authors choose an orthonormal operator basis and expand the operation elements in that basis.This construction yields the process matrix representation.Source: §II · p. 3
• Once the basis is fixed, the process matrix uniquely represents the quantum process rather than depending on a particular set of operation elements.This uniqueness makes it convenient for comparing processes.Source: §II · p. 3

Jamiolkowski representation

• The Jamiolkowski isomorphism maps a quantum operation E to a quantum state ρ_E, and this mapping is invertible.Consequently, knowing ρ_E is equivalent to knowing E, allowing state-based mathematical tools to analyze processes.Source: §II · p. 3

Representations

• The process matrix χ_E and Jamiolkowski state ρ_E are closely related and can be treated interchangeably for the paper’s purposes.The chosen operator basis gives χ_E = dρ_E as matrices.Source: §II · p. 3

Notation

• The notation ψ may denote either a pure state vector or its corresponding density matrix, with the intended meaning determined by context.For example, E(ψ) denotes applying a process to the density matrix associated with the pure state.Source: §II · p. 3

Comparison

• Trace distance and fidelity have complementary advantages, motivating the paper’s later development of process measures from both foundations.Fidelity-based metrics also inherit contractivity and convexity properties under the stated conditions.Source: §III · pp. 3–5

Trace distance

• The trace distance is a genuine metric whose physical meaning is the optimal distinguishability of two quantum states.For equally likely states, the optimal identification probability is 1/2 + D(ρ, σ)/2.Source: §III · p. 4
• Trace distance is contractive under quantum operations and doubly convex, supporting its use as a physically meaningful state-distance measure.Contractivity expresses that quantum processing cannot increase state distinguishability.Source: §III · p. 4

Fidelity

• For a pure target state, fidelity equals the overlap and the probability that an ideal test of the target state succeeds.For mixed states, the paper notes that no completely satisfactory physical interpretation is known.Source: §III · pp. 4–5
• Fidelity is a symmetric overlap measure bounded between 0 and 1, rather than a metric itself, and its square root is also called fidelity in some literature.The paper advocates the Eq. (4) definition because it later yields a probability-of-success interpretation for quantum computation.Source: §III · p. 4

Fidelity-based metrics

• The angle, Bures metric, and C(ρ, σ) convert fidelity into genuine metrics, with C introduced because it arises naturally in the later quantum-computation analysis.Fidelity-based metrics are monotonically decreasing functions of fidelity and satisfy the triangle inequality needed for chaining arguments.Source: §III · p. 4

Overview

• The paper’s goal is to recommend an error measure for comparing experimental quantum-information processes with theoretical ideals.The survey preliminarily assesses candidate definitions against criteria introduced earlier in the paper.Source: §IV · p. 5
• Process-error measures are organized into process-matrix, average-behavior, and worst-case approaches.Each approach is examined using both trace distance and fidelity.Source: §IV · p. 5
• The survey identifies four measures of particular merit and defers detailed discussion of their properties to the next section.The authors describe connections among the candidate measures while narrowing the broader set.Source: §IV · p. 5

Nomenclature

• The notation distinguishes state metrics from process metrics and separates fidelity-derived metrics from fidelity-based process measures.For example, Δ_ave denotes an average state-metric process measure, while F_ave denotes average fidelity.Source: §IV · p. 5

Definition and basic properties

• A process-matrix distance Δ_pro is formed by applying a quantum-state metric to the process matrices of two processes.This construction automatically gives the process measure the metric property.Source: A. Error measures based on the process… · p. 5

Stability and chaining

• Process-matrix measures satisfy stability when the underlying state metric is stable, while chaining requires contractivity and holds here when the ideal first process is doubly stochastic.Unitary ideal processes meet the doubly stochastic condition, covering the paper’s usual comparison case.Source: A. Error measures based on the process… · pp. 5–6

Physical interpretation

• The process-matrix measures D_pro and F_pro have operational interpretations through the average probability that a quantum computation fails or succeeds.This supplies the physical interpretation that remains to be established after the metric construction.Source: A. Error measures based on the process… · p. 6

Experimental accessibility

• For unitary ideal operations, F_pro can require at most 2d^2 observable averages, compared with d^4 − d^2 for full process tomography of D_pro.For qubits, the stated fidelity-based requirement can be d^2 observable averages.Source: A. Error measures based on the process… · p. 6
• A direct spanning-set method evaluates F_pro by choosing d^2 experimentally preparable input states and corresponding measurable observables.The required combinations depend on a known matrix determined by the inputs, observables, and ideal operation.Source: A. Error measures based on the process… · p. 6
• With a suitable choice of observables, F_pro can be evaluated using between d^2 and 2d^2 measurements, though the observables are then fixed by the ideal operation and chosen inputs.The method is generally less demanding than full process tomography and can simplify experimental-error estimation.Source: A. Error measures based on the process… · p. 6
• For n-qubit processes using Pauli-product bases and specified single-qubit inputs, F_pro requires d^2 observable averages instead of d^4 − d^2 for full process tomography.The selected observables remain Hermitian under this construction.Source: A. Error measures based on the process… · pp. 6–7

Limitations and open questions

• The minimum measurement count for F_pro under restricted inputs or observables remains an open problem, including the case of separable inputs and product observables.The paper identifies this as a direction for further exploration rather than a resolved optimum.Source: A. Error measures based on the process… · p. 7

Average-based measures

• Only the metric and chaining criteria hold generally for average-output measures, regardless of the underlying state metric.Both properties follow directly from the metric property of the state distance.Source: B. Error measures based on the average… · p. 7
• Average-based measures are difficult to calculate and measure experimentally because their defining integrals generally lack simple closed forms.No simple expressions are known for the discussed average distances.Source: B. Error measures based on the average… · p. 7

Physical interpretation

• Dave has a physical interpretation as the average bias in optimally distinguishing E(ψ) from F(ψ), but fidelity-based average measures lack a clear physical meaning.For uniformly random known inputs, the optimal distinguishing probability is 1/2 + Dave(E, F)/2.Source: B. Error measures based on the average… · p. 7

Stabilization

• Stabilizing the average distance with a large ancilla yields the process distance, ∆stab-ave(E, F) = ∆pro(E, F).The reduction follows from the near-maximal entanglement of randomly chosen states on a sufficiently large composite system.Source: B. Error measures based on the average… · p. 7

Average fidelity

• For unitary U, Fave(E, U) is easy to calculate and measure, but it is neither a metric nor stable.Any stable metric derived from Fave is equivalent in content to a function of Fpro.Source: B. Error measures based on the average… · p. 7

Conclusion

• The section concludes that none of the defined average-case error measures is particularly attractive, despite their natural motivation.The negative results reject plausible approaches before selecting preferred measures.Source: B. Error measures based on the average… · p. 7

Definition

• Worst-case distances maximize a state metric over pure inputs, and double convexity shows that mixed inputs cannot produce a larger value.The analogous statement for minimum fidelity follows from double concavity.Source: C. Error measures based on the worst… · p. 8

Stabilization

• The unstabilized worst-case distance Dmax is not stable, motivating stabilized measures formed by adjoining an identity-acted ancillary system.The stabilization procedure is extended to arbitrary state metrics and to minimum fidelity.Source: C. Error measures based on the worst… · p. 8

Physical interpretation

• Stabilized trace distance measures the worst-case distinguishing bias when arbitrary-size ancillas may assist discrimination.The fidelity-based stabilized measures receive related interpretations in quantum computation.Source: C. Error measures based on the worst… · p. 8

Calculation

• Dstab and Fstab, together with Astab, Bstab, and Cstab, can be calculated numerically by convex optimization.Convex optimization avoids false local minima because every local minimum is global.Source: C. Error measures based on the worst… · p. 8

Measurement

• Experimentally, stabilized measures require process tomography followed by numerical optimization, because general formulas are unavailable.For noisy processes compared with ideal unitaries, even single-qubit analytical formulas remain unsolved.Source: C. Error measures based on the worst… · p. 8

Candidate measures

• Dpro, Fpro, Dstab, and Fstab are superior to the other assessed error measures under the introduction’s criteria.For fidelity, the equivalent measures Cpro and Cstab are used for convenient interpretation.Source: §V · p. 8
• Their natural emergence in quantum-computation contexts makes these four measures the most deserving candidates for quantum-process error characterization.The paper defers the final choice among them to the conclusion.Source: §V · p. 9

Application structure

• The paper analyzes these measures separately for function computation and sampling computation, considering both worst-case and average-case performance.The two computational divisions arise naturally but are not generally reducible to one another.Source: §V · p. 9

Setup

• Function computation aims to produce f(x) with high probability from input instance x, as illustrated by Grover’s algorithm.The final computational-basis measurement should return f(x).Source: A. Function computation · p. 9

Function computation in the worst case

• For worst-case function computation, the actual error probability is bounded using Dstab(E, F) or Cstab(E, F) added to the ideal error probability.Which bound is tighter depends on the circumstances; when the ideal error is zero, the comparison depends on Dstab(E, F) versus Cstab(E, F)^2.Source: A. Function computation · p. 9
• When the ideal computation has zero error, either worst-case bound may be tighter because Dstab(E, F) can be larger or smaller than Cstab(E, F)^2.Both possibilities can occur under the relation between the two measures.Source: A. Function computation · p. 9

Function computation in the average case

• For average-case function computation, the average real-operation error is related to the ideal error through Dpro(E, F).The displayed relation is presented as an analogue of the worst-case bound for average inputs.Source: A. Function computation · p. 9
• The paper does not establish a full fidelity-based analogue of the average-case relation, proving only a partial result when the ideal computation succeeds with probability one.The corresponding condition is e_id = 0.Source: A. Function computation · p. 9

• Sampling computation targets an ideal outcome distribution p_x(y), while an imperfect process produces q_x(y) for the same problem instance.Source: B. Sampling computation · p. 10
• The Kolmogorov distance D(p,q) and Bhattacharya overlap F(p,q) are the classical distribution measures used for comparing real and ideal sampling outcomes.They are presented as commutative analogues of trace distance and fidelity.Source: B. Sampling computation · p. 10
• The Kolmogorov distance has an interpretation as distinguishability bias, while no similarly simple interpretation is known for the Bhattacharya overlap.Source: B. Sampling computation · p. 10
• In the worst case, the comparison is taken over problem instances x, whereas the average case compares joint distributions over uniformly sampled instances and outcomes.Source: B. Sampling computation · p. 10

VI. SUMMARY, RECOMMENDATIONS, AND CONCLUSION

• Four measures—J distance, J fidelity, S distance, and S fidelity—satisfy most or all of the proposed criteria and are identified as having particular merit.Source: §VI · p. 10
• All four measures satisfy stability and chaining properties, allowing total error in multistage tasks to be bounded from component processes.Source: §VI · pp. 10–11
• S distance and S fidelity bound worst-case error probabilities, whereas J distance and J fidelity provide bounds for average-case errors.Source: §VI · p. 11
• J fidelity is simplest to calculate and experimentally determine, but has the weakest operational interpretation and is mainly suited to early proof-of-principle demonstrations.Source: §VI · p. 11
• J distance handles intrinsic ideal computation errors but requires full process tomography, is harder to calculate, and remains tied to average errors.Source: §VI · p. 11
• S distance and S fidelity have stronger operational interpretations and are numerically tractable, but require full process tomography and lack elementary formulas.Source: §VI · p. 11
• The S distance and S fidelity are recommended as the two best bases for comparing real quantum-information experiments with theoretical ideals.Source: §VI · p. 11
• Because no convincing argument selects between the S measures, experiments should report both; further work includes simpler formulas, error bars, threshold connections, and non-trace-preserving operations.Source: §VI · p. 12

1. Proof of worst-case stabilization

• Worst-case stabilization is independent of ancillary-system dimension because a maximizing state needs at most d Schmidt coefficients.Source: §1 · p. 12

2. Proof of convex optimization property for Fstab

• Fstab computation reduces to minimizing a convex function over density matrices, while Dstab similarly reduces to maximizing a concave function over a convex set.Source: §2 · pp. 12–13

1. Function computation in the worst case

• For worst-case function computation, stabilized distance and fidelity bound the real error probability relative to the ideal error probability.Source: §1 · p. 13

2. Function computation in the average case

• Average-case function-computation bounds follow by analogous arguments, but the natural average-case analogue of one worst-case inequality is unavailable unless the ideal computation succeeds perfectly.Source: §2 · pp. 13–14

3. Sampling computation in the worst case

• For sampling computation, an imperfect quantum operation E is evaluated by how closely it reproduces the statistics of an ideal operation F acting on an input |x⟩.Source: §3 · p. 14