Fidelity Determines the Minimum Error of Quantum Recovery

The paper in detail

For a channel with Kraus operators \(A_j\), the worst-input entanglement fidelity is \(F(\Phi)=\min_\rho\sum_j|\operatorname{Tr}(\rho A_j)|^2\), the probability that a purification of \(\rho\) returns unchanged. Testing the ideal purified output gives \(D\ge1-F\). The square-root upper bound is attained by coherent errors, which is why fidelity and diamond error have long been treated as different scales. Qubit dephasing with probability \(p\) has \(F=1-p\) and \(D=p\) and is the extreme case.

The proof takes a worst input \(\rho_0\), its amplitudes \(\alpha_j=\operatorname{Tr}(\rho_0A_j)\), and the combination \(T=\sum_j\bar\alpha_jA_j\) with polar decomposition \(T=U^\dagger|T|\). Optimality of \(\rho_0\) forces \(\operatorname{Re}T\succeq fI\), so \(|T|\succeq fI\). Applying \(U\) and centring every Kraus operator on its scalar amplitude leaves a remainder of size at most \(1-f\) in diamond norm. The bound holds exactly at every noise strength, with no small-error expansion.

A fixed channel may admit a better correction than the guaranteed one. Optimising the coherent weights gives the best final unitary, and relaxing the weight vector to a density matrix makes this a semidefinite program whose optimum is rank one whenever its value exceeds one half. Fidelity-optimal and operationally optimal gates form nested families, with radii scaling as \(\sqrt\epsilon\) and \(\epsilon\). Under a residual unitary error of size \(s\), the polar correction loses only \(O(s^2)\), whereas some other operationally optimal gates lose \(O(s^{4/3})\).

The example is a doubly controlled phase error: on the control branch \(|11\rangle\), a flag fires with probability \(p\) and applies \(S=\operatorname{diag}(1,i)\) to the target. The optimal recovery over all channels is the controlled rotation \(e^{i\alpha_*Q\otimes Z}\) with \(\alpha_*=\tfrac12\arcsin p\). Average-fidelity calibration weights the active branch by only one quarter and under-rotates. Independent parallel blocks with local errors below one half are optimally corrected by the product of local polar corrections, even among entangling unitaries.

Transferable methods

1. Centring the Kraus operators on the worst-case amplitude

   The diamond norm of a near-identity channel is hard to bound directly, because the coherent part of the error enters linearly. Rotating by the polar part of the worst-input amplitude and subtracting each Kraus operator's scalar component leaves an exactly computable remainder whose size is the infidelity.

   Let \(f=F(\Phi)>0\) and \(T=\sum_j\bar\alpha_jA_j=U^\dagger|T|\). Editorial derivation: along \(\rho_0+t(\sigma-\rho_0)\), the objective \(\sum_j|\operatorname{Tr}(\rho A_j)|^2\) has derivative \(2\bigl(\operatorname{Re}\operatorname{Tr}(\sigma T)-f\bigr)\) at \(t=0\), which must be non-negative at a minimum. So \(\operatorname{Re}T\succeq fI\), and for unit \(v\), \(\|Tv\|\ge\operatorname{Re}\langle v,Tv\rangle\ge f\), giving \(|T|\succeq fI\).

   Set \(L_j=UA_j-\alpha_jI\). Using \(\sum_jA_j^\dagger A_j=I\) and \(UT=|T|\), \[H=\sum_jL_j^\dagger L_j=(1+f)I-2|T|,\qquad0\preceq H\preceq(1-f)I,\] and the corrected channel satisfies \((\mathrm{Ad}_U\Phi-\mathrm{id})(X)=\sum_jL_jXL_j^\dagger-\tfrac12(HX+XH)\). Editorial check: the cross terms equal \(KX+XK\) with \(K=|T|-fI\), and \(-\tfrac12\{H,X\}=-(1-f)X+KX+XK\), so both sides agree.

   The completely positive part has diamond norm \(\|H\|_\infty\), and the anticommutator term is at most the same, so \(D(\mathrm{Ad}_U\Phi)\le\|H\|_\infty\le1-f\). Editorial checks with a semidefinite-program diamond norm: dephasing gave \(F=1-p\) and \(D=p\) at \(p=0.05\), 0.2 and 0.45, and the chain \(1-F^\prime\le D^\prime\le1-F\le D\) held for 12 random qubit channels with coherent rotations, each with the polar correction computed as above.

   How to reuse it

   For any near-identity channel with coherent error, centre on the best-matching scalar and polar-correct. The infidelity then bounds the worst-case error with constant one rather than through a square root. This is useful wherever fidelity estimates are cheap and diamond-norm guarantees are needed, as in fault-tolerance thresholds.

2. A semidefinite program for the best final unitary

   The guaranteed correction uses the worst input's weights, which need not be optimal for a given channel. Optimising over all coherent weights is non-convex. Replacing the weight vector by a density matrix makes it a semidefinite program, and above one half the relaxation is tight.

   For normalised weights \(c\), let \(B_c=\sum_j\bar c_jA_j=U_c^\dagger|B_c|\) and \(q(c)=s_{\min}(B_c)^2\). Then \[\max_cq(c)=\max_UF(\mathrm{Ad}_U\Phi)=1-\min_UD(\mathrm{Ad}_U\Phi).\] The relaxation is \[\Gamma=\max\Bigl\{t:\ \tau\succeq0,\ \operatorname{Tr}\tau=1,\ \textstyle\sum_{ij}\tau_{ij}A_i^\dagger A_j\succeq tI\Bigr\}.\]

   If \(\Gamma>1/2\), take \(\lambda=\lambda_{\max}(\tau)\) with eigenvector \(c\). From \(\tau\preceq(1-\lambda)I+(2\lambda-1)cc^\dagger\), applying the unital map gives \[q(c)\ge t+\frac{(1-\lambda)(2t-1)}{2\lambda-1},\] so any optimum with \(\lambda<1\) could be improved. Every optimal \(\tau\) is therefore rank one, and its eigenvector gives the optimal final unitary \(U_c\), with error certified as \(1-\Gamma\).

   Editorial check of the paper's example at \(p=0.01\): the closed form \(D_*=\bigl(1-\sqrt{(1-p)/(1+p)}\bigr)/2=0.0049752\) matched an SDP diamond-norm computation for \(U_{\alpha_*}\). The uncorrected error came out as 0.007071 and the corrected worst-input fidelity as 0.99502475, both as reported. For the input \(|11+\rangle\) and a \(Y\) measurement on the target, \(\langle Y\rangle=-\operatorname{Im}\bigl[(1-p-ip)e^{2i\alpha}\bigr]\) drops from 0.01 without correction to 0.0000995 at \(\alpha_*\).

   How to reuse it

   For gate calibration, compute the final correction from the characterised channel with this SDP rather than maximising average gate fidelity, which underweights branches that a computation can occupy almost entirely. A value of \(\Gamma\) above one half certifies the result as optimal.

Source: Main text, "Construction and proof" (Eqs. 4-7), "Computing the best final unitary" (Eq. 9) and End Matter Appendix B. Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Fidelity optimisation, which is convex and well understood, already determines the best achievable operational error, and one final unitary makes a fidelity-optimal decoder attain it. This answers the open question of Holdsworth, Singh and Wilde without shared randomness. In practice it turns infidelity budgets into diamond-norm budgets that add up across a circuit. It also shows that calibrating gates by average fidelity can leave avoidable coherent error on the branches a computation actually uses.

Limitations. The results are for finite dimensions and a fully characterised channel; the correction depends on a worst-case input found by optimisation. The SDP is exact only when the best final-unitary error is below one half, and a hierarchy handles the rest. The three-qubit example is a model, and its implementation-noise comparison is in the supplement. Several proofs, including the singular-source case, the uniqueness of the average-fidelity comparator and the extension to isometries, are in Supplemental Material that the entry did not check. The author states that generative AI assisted with research and manuscript preparation, and that they directed the work and verified the outputs.

Description based on the full arXiv HTML of v1, read 2026-11-06: the main text and End Matter Appendices A to D, plus the acknowledgements' AI statement. The Supplemental Material, included in the same HTML, was not read.