Quantum computing · Theory
The Impact of Markovian Errors on Random Circuits
arXiv:2610.31500v1 ↗PDF ↗Submitted 22 October 2026
- Question
- Randomized benchmarking reports one error rate, but under gate-dependent noise it is unclear what that number measures. Average gate infidelity is gauge dependent, so it cannot be the answer. For small Markovian errors, how does each elementary error (Pauli-stochastic, over-rotation, axis tilt) enter the RB decay?
- Result
- For any gate set and sampling distribution forming a sequence-asymptotic unitary 2-design, Pauli-stochastic rates are the only errors that enter the decay linearly. Hamiltonian rates enter through a quadratic form that vanishes on gauge directions. For direct RB of \(\{X_{\pi/2},Y_{\pi/2}\}\) with probability \(p\) of \(X\), the form is explicit in two over-rotations and one relative axis tilt. Equal over-rotations \(\theta\) give \(r=3\theta^2/4\) at \(p=1/2\), three times each gate’s infidelity, and about 16.6 times at \(p=0.9\). For Clifford RB, a fixed form with weights \(1/9\) and \(1/18\) is filtered through the compilation.
The paper in detail
The motivating example has native gates \(X_{\pi/2}\) and \(Y_{\pi/2}\) with over-rotations \(\theta_X,\theta_Y\) and a tilt \(\phi\) of the angle between their axes. Four error models are chosen with the same leading native-gate entanglement infidelity. Over 2000 random compilations of the 24 single-qubit Cliffords, the Clifford-RB rate per native gate varies widely and differs between models. In direct RB, which samples native gates with probability \(p\) for \(X\), the weighted-infidelity heuristic predicts flat curves, but the actual rates depend strongly on \(p\) and differ between models.
Each noisy gate is written as \(\widetilde{\mathcal G}_k=e^{L_k}\mathcal G_k\), with \(L_k\) expanded in Hamiltonian, stochastic, correlation and active generators. A small gauge change \(e^K\) shifts \(L_k\) by \(K-\mathcal G_kK\mathcal G_k^{-1}\), and first-order gauge-invariant (FOGI) coordinates are the combinations unchanged by every such shift. The RB decay is the eigenvalue \(\gamma\) of \(\mathscr L=\sum_k\Omega_k\mathcal G_k\otimes\widetilde{\mathcal G}_k\) continuing the ideal depolarizing mode. A spectral gap of the ideal moment operator, which defines a sequence-asymptotic 2-design, keeps it isolated for small errors, and \(r=\tfrac{d^2-1}{d^2}(1-\gamma)\).
Theorem 1 states \[\gamma=1-\tfrac{d^2}{d^2-1}\sum_{k,P}\Omega_ks_P^{(k)}-Q_H(\mathbf h;\Omega)+O(\delta^2,\delta\theta,\theta^3).\] The linear term follows from degenerate perturbation theory on the two-dimensional ideal eigenspace: only the superoperator trace of the averaged generator matters, and only stochastic generators have nonzero trace. The quadratic term is gauge invariant because \(\gamma\) is exactly invariant under similarity transformations and has no linear Hamiltonian part. Therefore \(Q_H\) annihilates every linearized gauge direction and depends only on FOGI coordinates.
For \(\{X_{\pi/2},Y_{\pi/2}\}\), the \(6\times6\) Hamiltonian form has rank 3 with the gauge directions as kernel. Its reduced form in \(F_1\approx\theta_X/2\), \(F_2\approx\theta_Y/2\) and the relational \(F_3\approx\phi/2\) is \(\mathbf Q_{\rm DRB}(p)\). The tilt weight \(16pq/3\) vanishes when either gate disappears. The intrinsic weights diverge like \(1/q\) and \(1/p\) because consecutive copies of the same gate add their errors coherently. For Clifford RB, first-order compiled vectors \(\mathbf h_C\) give \(Q^{(2)}_{\rm CRB}=\tfrac1{18}\sum_C\|\mathbf h_C\|^2+\tfrac1{216}\mathbf u\cdot\mathbf w\), equivalent to weight \(1/9\) on three symmetric directions, \(1/18\) on the other 66 and zero on gauge. Compilation changes how native errors project onto these sectors.
Transferable methods
First-order RB decay from a Choi-space projector
The unperturbed eigenvalue-one space of the RB moment operator is spanned by the identity and the completely depolarizing map, both fixed by every ideal gate. First-order shifts reduce to projecting the averaged error generator onto this space, which depends only on two traces.
With \(\widetilde{\mathcal G}_k=(1+\varepsilon g_k)\mathcal G_k\), every \(X\in\mathcal V_1=\operatorname{Span}\{\Delta,\mathrm{id}\}\) commutes with ideal unitary channels, so \(\widehat{\mathscr L}_1(X)=\bar g\circ X\) with \(\bar g=\sum_k\Omega_kg_k\). The Gram matrix of \(\{\Delta,\mathrm{id}\}\) in the Choi inner product is \(\begin{pmatrix}1&1\\1&d^2\end{pmatrix}\), which gives \(P_{\mathcal V_1}(\Phi)=\tfrac{d\operatorname{tr}\Phi(I)-\operatorname{Tr}\Phi}{d^2-1}\Delta+\tfrac{\operatorname{Tr}\Phi-\operatorname{tr}\Phi(I)/d}{d^2-1}\mathrm{id}\).
Trace-preserving generators have \(\operatorname{tr}\bar g(I)=0\), so \(P_{\mathcal V_1}(\bar g\circ D)=\tfrac{\operatorname{Tr}\bar g}{d^2-1}D\) with \(D=\mathrm{id}-\Delta\), and \(\delta\gamma=\operatorname{Tr}\bar g/(d^2-1)\). Since \(\operatorname{Tr}H_P=\operatorname{Tr}C_{P,Q}=\operatorname{Tr}A_{P,Q}=0\) and \(\operatorname{Tr}S_P=\operatorname{tr}(P)^2-d^2=-d^2\), the result is \(r=\sum_k\Omega_k\sum_Ps_P^{(k)}+O(\delta^2,\delta\theta,\theta^2)\).
Editorial remark: for stochastic noise this is the sampling-weighted sum of Pauli error rates, so the naive heuristic is right at first order. Its failure for coherent errors in Figs. 1–2 is entirely a second-order effect.
How to reuse it
Any protocol whose signal decays as powers of a twirled moment operator (RB variants, cycle benchmarking, mirror circuits) can be analysed this way: identify the ideal eigenvalue-one space, build its projector from Choi pairings, and read the first-order shift from traces. It needs a spectral gap of the ideal moment operator, and the constants worsen as that gap closes.
Gauge invariance forces the coherent term onto FOGI coordinates
Since \(\gamma\) is an exact gauge invariant with no linear Hamiltonian term, its quadratic Hamiltonian coefficient must vanish on every linearized gauge direction. The form then lives on the FOGI quotient, which for the example gate set is three-dimensional.
Set \(\mathbf h=\varepsilon\mathbf a\) and gauge by \(e^{\varepsilon tK}\), so \(\mathbf h'=\varepsilon(\mathbf a+t\mathbf g_H(K))+O(\varepsilon^2)\). The \(O(\varepsilon^2)\) correction affects \(\gamma\) only at order \(\varepsilon^3\) because there is no linear term. Comparing \(\varepsilon^2\) coefficients of \(\gamma(\mathbf h')=\gamma(\mathbf h)\) gives \(Q_H(\mathbf a+t\mathbf g)=Q_H(\mathbf a)\) for all \(t\), so \(\mathbf a^{\mathsf T}\mathbf Q_H\mathbf g=0\) for all \(\mathbf a\) and \(\mathbf Q_H\mathbf g_H(K)=0\).
At \(p=1/2\), \(\mathbf Q_{\rm DRB}=\tfrac13\begin{pmatrix}5&1&2\\1&5&2\\2&2&4\end{pmatrix}\), with eigenvalues \(8/3\) (equal \(F_i\)), \(4/3\) (differential over-rotation) and \(2/3\) (direction \((-1,-1,2)\)). For equal over-rotations \(\mathbf F=(\theta/2,\theta/2,0)\), \(\mathbf F^{\mathsf T}\mathbf Q\mathbf F=\tfrac{\theta^2}{4}\cdot\tfrac{5+5+2}{3}=\theta^2\), so \(r=\tfrac34\theta^2\).
Editorial check: building the \(9\times9\) direct-RB \(\mathscr L\) matrix from the exact noisy rotations of Eqs. (3)–(4) with \(\theta=0.004\), the depolarizing eigenvalue matches Eqs. (80)–(82) to within 0.3% at \(p=0.3\), 0.5 and 0.9 for the equal, opposite and tilt models.
How to reuse it
For any gauge-invariant figure of merit whose linear coherent term vanishes, the leading coherent contribution must be a form on FOGI coordinates. Find the gauge kernel numerically, then report sensitivities to intrinsic and relational errors separately. A single scalar cannot fix error signs, so combine several sampling settings or deliberate error injection.
Context & limitations
Why it matters. It explains why direct-RB rates depend on the sampling distribution and Clifford-RB rates on the compilation, and it turns an RB number into a function of physical, gauge-invariant error parameters. Running at several sampling probabilities gives quadratic constraints on coherent errors.
Limitations. The expansion is perturbative at fixed sampling. Its coefficients diverge as \(p\to0\) or \(1\), and it fails first there. Explicit forms are given only for one single-qubit gate set. Correlation and active generators are treated only at leading order, and multi-qubit gate sets are left open. Quadratic constraints cannot fix the signs of errors. Drift, leakage and non-Markovian noise are excluded.