Sharpening Worst-Case Error Assessment for Fault-Tolerant Quantum Computing: Fidelity and Its Deviation

TL;DR

It is shown that the fidelity and its deviation can provide an economical, operationally meaningful, and accurate standard for assessing the fault tolerance of the engineered quantum gates and circuits.

Abstract

Gate fidelity -- an average fidelity over all possible input states -- is the workhorse metric for benchmarking quantum gates or circuits, yet fault-tolerant quantum computing ultimately depends on the worst-case behavior, typically quantifiable by so-called the diamond distance. In the low-error regime, the coherent errors can inflate the worst-case error even when the reported gate fidelity is high, making the gate fidelity alone an unreliable proxy for fault-tolerance readiness. To capture the missing information, we introduce a companion observable -- what we dub the fidelity deviation -- that quantifies how strongly the state-dependent fidelities fluctuate across input states. Adopting such fluctuations in assessing the fault-tolerance is physically natural because some input directions are nearly unaffected while others form narrow "valleys" that dominate adversarial circuit behavior. For coherent (unitary) gate errors on two or more qubits, we show that the gate fidelity together with the fidelity deviation constrains the relevant spectral moments of the error unitary, enabling an explicit and tight certificate of the worst-case error. Both quantities are estimated directly from the same randomized input-measurement experiment, without full process tomography. We show that the fidelity and its deviation can provide an economical, operationally meaningful, and accurate standard for assessing the fault tolerance of the engineered quantum gates and circuits.

Figures (7)

• Figure 1: Fluctuation-assisted worst-case certification from $(F,D)$.(a) Concept The quality of a quantum gate should be treated as a fidelity landscape rather than a single number. From the interaction-picture error channel $\mathcal{E}$, one samples the state-dependent single fidelities $f_{\mathcal{E}}(\psi)$ over randomized input states $\left|\psi\right>$. The mean of this distribution is the gate fidelity $F$, while its width is the fidelity deviation $D$. (b) Theory. In the coherent (unitary) regime, $D$ remains informative even when unitarity saturates. For two-qubit and larger coherent errors, the pair $(F, D)$ fixes the spectral moment invariants of the effective error unitary and yields an explicit certified overlap $c(F, D)$ ( Theorem \ref{['thm:main']}), which directly translates into a tight worst-case certificate on the diamond distance. (c) Experiment. Both $F$ and $D$ are extracted from the same randomized input-measurement experiment: one prepares random states (for example from a unitary design), applies the implemented gate followed by the ideal inverse, and performs a projective "identity" test. A simple factorial-moment correction removes the bias from finite-shot (binomial) noise when estimating the second moment, enabling an unbiased estimate of $D$ from the same data stream used for $F$ (Methods). (d) Regime adaptivity. When the incoherence dominates, the unitarity-based bounds can certify linear-in-infidelity worst-case behavior; when the coherence dominates and unitarity saturates, the fluctuation information in $D$ restores high-resolution worst-case certification.
• Figure 2: Three coherent examples and consolidated worst-case certification. Single-qubit primitives carry a small coherent over-rotation about their native unitary axis (for example, a calibration error on the generator of a $T$ or Hadamard gate). Two-qubit primitives carry a coherent miscalibration, such as, a controlled-phase over-rotation or a controlled-$X$ over-rotation on a CNOT-like interaction. Here, we consider three examples: (a) a CZ-like entangling gate with a coherent phase error on the $\left|11\right>$ component, (b) a standard Clifford+$T$ decomposition of the Toffoli gate, where each primitive (single-qubit gates and CNOTs) is affected by the same coherent over-rotation parameter, and (c) the $10$-qubit QFT circuit built from Hadamard and controlled-phase gates, with a uniform coherent over-rotation applied to each primitive. For each example, we compare the exact diamond distance of the resulting coherent error (solid line) with three computable upper bounds obtained from experimentally accessible data: the fidelity-only conversion bound, Kueng et al.'s $(r,u)$-based bound evaluated in the coherent limit, and our proposed $(F, D)$-assisted bound from Theorem \ref{['thm:main']}. All plots use a logarithmic vertical axis to highlight the low-error regime relevant to fault tolerance. Across all three examples, incorporating the fidelity deviation $D$ yields a substantially tighter worst-case certificate in the coherent regime, while the unitarity-assisted bound becomes increasingly loose with system size because unitarity necessarily saturates for unitary errors.
• Figure 3: CZ-like coherent phase error: exact diamond distance vs. upper bounds, with protocol-based estimates. We consider the unitary error $\hat{X}=\mathrm{diag}(1,1,1,e^{i\phi_{\epsilon}})$ arising from a coherent phase miscalibration of a controlled-phase gate. The solid curve shows the exact worst-case error $d_{\diamond}(\mathcal{X},\mathcal{I})=\left|\sin(\phi_{\epsilon}/2)\right|$ from Eq. (\ref{['eq:cz_diamond_exact_main']}). The dashed curve shows the fidelity-only bound $\sqrt{20r}$ at $d=4$. The dotted curve shows Kueng's $(r,u)$-based upper bound in Eq. (\ref{['eq:kueng_ru_bound_recalled_main']}) evaluated at $u=1$, which is provably looser than $\sqrt{20r}$ by the factor $2\sqrt{2}$ in this coherent regime. The dash-dotted curve shows our moment-assisted bound $\sqrt{1-c(F,D)^2}$ from Theorem \ref{['thm:main_d4']}, which substantially sharpens worst-case certification by exploiting the fluctuation information encoded in $D$. In addition, the markers overlay finite-sample, protocol-based estimates obtained by simulating the direct $(F, D)$ estimation procedure of Sec. \ref{['sec:FD_measurement']}: for each $\phi_{\epsilon}$ we estimate $(\hat{F}, \hat{D})$ from $M=500$ random input states and $N=1000$ projective-test shots per state, and then evaluate the $F$-only and $(F,D)$-assisted bounds at $(\hat{F},\hat{D})$ as in Eq. (\ref{['eq:cz_protocol_bounds_main']}). The vertical axis is shown on a logarithmic scale to highlight the separation between average-case and worst-case estimates in the small-error regime relevant to fault tolerance.
• Figure 4: A standard decomposition of Toffoli gate.$6$ CNOTs, $2$ Hadamard, and $7$$\hat{T}$ (or $\hat{T}^\dagger$) amy2013meet.
• Figure 5: Decomposed Toffoli with coherent primitive over-rotations: true diamond distance vs. upper bounds. We consider a three-qubit Toffoli gate implemented by the standard $6$-CNOT Clifford+$T$ decomposition as in Eq. (\ref{['eq:toffoli_decomposition_ideal']}) and/or Fig. \ref{['fig:Toffoli_decomp']}. Every primitive (all CNOTs and all single-qubit gates) is assumed to carry the same coherent over-rotation parameter $\epsilon$ according to Eqs. (\ref{['eq:T_overrot_model']})--(\ref{['eq:CNOT_overrot_model']}), yielding a unitary error channel with $u=1$. The solid curve shows the true diamond distance computed from the convex-hull characterization in Eq. (\ref{['eq:toffoli_m_from_convex_hull']}). The dashed curve shows the fidelity-only bound $\sqrt{72r}$. The dotted curve shows Kueng's $(r, u)$-based upper bound evaluated at $u=1$, which is amplified by the factor $8/\sqrt{2}$ and becomes extremely loose in this coherent regime. The dash-dotted curve shows our $(F,D)$-assisted bound $\sqrt{1-c_{8}(F,D)^{2}}$, which uses the fluctuation information encoded in $D$ and yields substantially sharper worst-case certification. The vertical axis is logarithmic to emphasize separation in the low-error regime.

Theorems & Definitions (44)

• Lemma 1: Diamond distance for unitary errors
• proof : Proof sketch.
• Lemma 2: Spectral invariants from $(F,D)$ for $d\ge 4$
• proof : Proof sketch.
• Theorem 1: Moment-assisted coherent certificate for $d\ge 4$
• proof : Proof sketch.
• Theorem 2: Hybrid regime-adaptive certification
• Proposition 1: Stability under idle ancillas
• proof
• Proposition 2: Chaining / subadditivity under composition