Bounding the average gate fidelity of composite channels using the unitarity

• Randomized benchmarking can precisely estimate average gate fidelity under robust, experimentally realistic conditions in a physical operational gauge [15–24].Source: §I · p. 1
• Interleaved randomized benchmarking estimates individual gate infidelities but suffers a systematic error that can significantly limit precision.Source: §I · p. 1
• Component gates with identical fidelity can produce dramatically different circuit errors because stochastic errors accumulate linearly while unitary errors accumulate quadratically at low infidelity.Source: §I · p. 1
• Unitarity-based information yields tighter bounds that interpolate between quadratic coherent-error growth and linear stochastic-error growth, while improving precision for interleaved benchmarking.Source: §I · pp. 1–2
• The paper first derives optimal circuit-infidelity bounds from component infidelities alone, which are saturated by unitary channels and therefore cannot be improved without additional error information.Source: §I · p. 1

• Markovian quantum processes are modeled as completely-positive trace-preserving maps on density operators, with channel composition written multiplicatively.Source: §II · p. 2
• Quantum channels are represented using Kraus operators, χ matrices, and Liouville transfer matrices relative to a trace-orthonormal operator basis.Source: §II · p. 2
• The Liouville representation separates a channel into a non-unital vector and unital block, from which unitarity and the effective depolarizing constant are expressed.Source: §II · pp. 2–3
• The effective depolarizing constant p and χ00 are linear functions of fidelity, with their relations to fidelity and infidelity tabulated in Table I.Source: §II · p. 3, Table I · p. 3

• Strict upper and lower bounds on composite infidelity are obtained from component infidelities, and unitary processes saturate them for all even-dimensional systems.Source: §III · p. 3
• The two-channel bound strictly improves Ref., is saturated for all relevant variable values, and is therefore optimal when only component infidelities are known.Source: §III · p. 3
• For m equal-infidelity channels, the resulting upper bound displays the expected quadratic scaling in m and cannot be improved without additional error information.Source: §III · p. 3
• The bounds are tight because there exist channel pairs saturating both inequality signs, with explicit unitary constructions establishing saturation in even dimensions.Source: §III · pp. 3–4
• An inductive extension gives a bound for compositions of m channels that inherits the tightness of the two-channel result.Source: §III · p. 4
• The worst possible composite fidelity arises through coherent, unitary buildup, reflected by the trigonometric form of the bound.Source: §III · p. 4

Unitarity and coherence angle

• The unitarity quantifies the intermediate regime between Pauli and unitary errors, with θ(E)=0 for depolarizing channels, θ(E)=O(r) for Pauli channels, and θ(E)=O(√r) for unitary channels.It is bounded by θ(E)∈[0, arccos p(E)], and reaches its maximum when the channel is unitary.Source: §IV · p. 4

Unitarity-based bounds

• Combining infidelity and coherence angle yields improved bounds on composite infidelity for m unital channels or m single-qubit operations with equal r and θ.The resulting expression includes higher-order terms in r and θ^2 and is saturated to the appropriate order in even dimensions.Source: §IV · pp. 4–5

Pauli and unitary limits

• For Pauli errors, θ^2=O(r^2), so the unitarity-based bound recovers linear growth in the number of components to lowest order.This recovers the Pauli-channel scaling rather than the quadratic worst-case behavior allowed by infidelity-only bounds.Source: §IV · pp. 4–5
• For unitary errors, (d−1)θ^2=2dr+O(r^2), so the same bound recovers the quadratic infidelity-growth regime.Thus the coherence-angle bound connects the best-case Pauli behavior to the worst-case unitary behavior.Source: §IV · p. 4

Long compositions

• Theorem 4 bounds the effective depolarizing constant for m channels with common p and u, with a tighter version for unital channels.Its binomial factor reveals quadratic dependence on m, while the deviation from the naive value pm grows with the coherence angle.Source: §IV · p. 5

Pauli-channel improvement

• When one channel is Pauli, the composite infidelity is essentially linear in the individual infidelities because Pauli twirling converts the other channel into a low-coherence Pauli channel.This improves over directly applying the general two-channel bound.Source: §IV · p. 5

Motivation

• Interleaved RB estimates F(EhE), the composition of the target-gate error and gate-set error, rather than F(Eh) directly.The resulting bounds on F(Eh) can have a wide range because possible coherent effects cannot be ignored.Source: §V · p. 5
• The same interleaved-RB data can correspond to target-gate fidelities of 0.9991 or 0.9975, showing that unitarity-blind estimates may be loose.Both simulations fix F(E) = 0.9975 and F(EhE) = 0.9960, but differ in whether the target error is highly coherent or depolarizing.Source: §V · pp. 5–6

Baseline bound

• For a 12-element Clifford subgroup forming a unitary 2-design, the naive individual-gate bound is at worst a factor of 3/2 worse than the interleaved-benchmark bound.This comparison uses a bound requiring no additional experiment.Source: §V · p. 6

Unitarity-enhanced bound

• Unitarity information yields bounds on F(Eh) that can be orders of magnitude tighter than estimates based on interleaved RB alone.If the errors were depolarizing, F(Eh) could be exactly estimated from interleaved RB; generally, unitarity quantifies departure from that model.Source: §V · p. 6

Experimental procedure

• The recommended procedure combines standard RB for p(E), unitarity RB for u(E), and interleaved RB for p(EhE), then applies eq. (42) and table I.The analysis assumes a gate-independent noise approximation for the URB component.Source: §V · p. 6

Interpretation

• The paper also notes that the depolarizing identity u(E) = p(E)^2 makes eq. (42) reduce to p(Eh) = p(EhE)/p(E), while stochastic Pauli errors satisfy this through order r(E)^2.Treating infidelity as linear under composition is reliable only when errors are mostly stochastic, because coherent effects can also arise from faulty control.Source: §V · p. 6

• Interleaved randomized benchmarking simulations compare a coherent high-fidelity individual error, F(Eh) = 0.9991, with a stochastic error, F(Eh) = 0.9975, under the same average error E of fidelity 0.9975.The figure plots average survival probability against sequence length for the two error models.Source: Figure 1 · p. 7
• Quadratic infidelity growth for coherent unitary errors contrasts with linear growth for stochastic Pauli channels, limiting what component fidelities alone reveal about circuits.This disparity means gate-fidelity characterization alone supports only weak statements about larger circuit constructions.Source: §VI · pp. 6–7
• A coherence angle captures rotation-like error components and enables tighter circuit-error bounds that smoothly interpolate between linear and quadratic scaling.The angle is contrasted with contraction on the Bloch space and is designed to use robustly estimable quantities.Source: §VI · p. 7
• The new bound substantially improves individual-gate fidelity estimates from interleaved randomized benchmarking, but its practicality assumes closely related gauges for RB-derived unitarities and average fidelities.Relaxing this gauge assumption is identified as an open problem.Source: §VI · p. 7
• Unitarity-dependent bounds tighten estimates of an individual gate fidelity from the composite fidelity F(EhE), with randomly generated channels evaluated at fixed F(E) = 0.9975.The bounds are shown across u(E) = 1.00000, 0.99300, 0.99030, and 0.99003 ≈ p(E)^2.Source: Figure 2 · p. 8
• Applying the bound to experimental two-qubit data yields infidelity bounds spanning up to two orders of magnitude under maximal coherence, while the stochastic assumption gives narrower bounds.The stochastic model assumes u(E) ≈ p(E)^2, and statistical uncertainty produces nonzero error bars.Source: Figure 3 · p. 8

• The appendix defines a coherence angle for real matrices and uses it to organize matrix-product inequalities.Source: Appendix A · p. 9
• Both bounds are saturated for all listed norms and coherence angles in even dimensions, establishing tightness of the inequalities.Source: Appendix A · p. 9
• Cauchy–Schwarz bounds the centered trace product by ∥M1∥F ∥M2∥F sin θ(M1) sin θ(M2).Source: Appendix A · p. 9
• Trace identities rewrite the centered product as |Tr M1M2 − d^−1 Tr M1 Tr M2|, which combines with trigonometric identities to yield both inequalities.Source: Appendix A · p. 9
• The lower bound is generalized from two matrices to products M1:m = M1M2 · · · Mm.Source: Appendix A · p. 9