Mirror Circuit Fidelity Estimation

• Mirror circuit fidelity estimation is a technique that constructs quantum circuits with a mirrored structure to robustly measure process infidelity and coherence properties.
• It employs randomized benchmarking and Pauli twirling to extract exponential decay rates, linking them directly to average error per layer while mitigating SPAM effects.
• Scalable protocols like MRB enable efficient benchmarking for systems with tens to hundreds of qubits, providing clear hardware performance metrics and error bounds.

Mirror circuit fidelity estimation refers to a family of protocols for measuring the fidelity—or, equivalently, the infidelity—of quantum circuits and devices by leveraging “mirrored” circuit structures, in which a random quantum circuit is combined with its inverse. This approach, realized by mirror benchmarking methods such as mirror-circuit randomized benchmarking (MRB), provides scalable, robust, and system-level estimates of multi-qubit logic layer performance, in particular by extracting exponential decay rates under noise and relating these to average process fidelities, infidelities, and, in some cases, the coherence of the noise. These protocols are engineered to overcome the scalability bottlenecks of standard randomized benchmarking and to furnish direct, device-level performance curves, including for systems with tens or hundreds of qubits (Proctor et al., 2021, Mayer et al., 2021).

1. Mirror Circuit Construction and Protocols

Mirror-circuit fidelity estimation consists of preparing quantum circuits $C$ composed of a random sequence of logic layers drawn from a gate set $G$, followed by the exact inverse of these operations in reversed order, yielding a "mirror" structure. Formally, for a sequence of $L$ random layers $g_1, \ldots, g_L$, the mirror circuit is

$$C = g_1 g_2 \cdots g_L g_L^{-1} \cdots g_2^{-1} g_1^{-1}$$

Random Pauli layers may be interleaved between $g_i$ to “twirl” the noise and realize randomized compiling, enhancing noise symmetrization (Proctor et al., 2021). The input is typically a computational basis state, and the output is measured in the computational basis; in the noiseless case, the final measured bit-string matches the initial state exactly.

An essential distinction in MRB protocols is the use of Clifford logic layers and Pauli dressing: each round samples and applies a random one-qubit Clifford $F_0$ to each qubit, and then alternates sampled Pauli layers $P_i$ and sampled Clifford logic layers $L_i \sim \Omega$, ensuring that $\Omega$ scrambles errors (local twirl + spread), is inversion symmetric, and that $L^{-1}$ is in the layer set $\mathcal{L}$. The mirror property guarantees that, in the absence of noise, the final state is always deterministic and known classically.

2. Survival Probability, Exponential Decay, and Effective Polarization

The principal observable in mirror benchmarking is the average survival probability $S(L)$, defined as the probability of measuring the output state that matches the ideal mirror circuit outcome. In practice, $S(L)$ is estimated by the following protocol:

• For each depth $d$ (mirror sequence length), sample $K$ random mirror circuits and perform $N$ shots per circuit.
• Given the probability $h_k(C)$ to observe $k$ bit flips from the expected output, define the “effective polarization” for circuit $C$ as

$$S(C) = \frac{4^n}{4^n-1} \sum_{k=0}^{n} \left(-\frac{1}{2}\right)^k h_k - \frac{1}{4^n-1}$$

Averaging over $K$ circuits gives the mean polarization $\overline{S}_d$ for depth $d$.

• $\overline{S}_d$ is then fit by an exponential model

$\overline{S}_d \approx A p^{d}$

where $A$ is a SPAM-dependent prefactor and $p$ is the decay parameter characterizing average circuit coherence.

This exponential decay is a general consequence of the uniform noise assumption and sufficient twirling via a group forming a unitary 2-design (usually the $n$-qubit Clifford group or a major subset). Under these circumstances, the decay rate $p$ is determined by the unitarity of the noise channel (see Section 4).

3. Extracting Average Infidelity and Fidelity Bounds

In randomized benchmarking and mirror benchmarking, the exponential decay parameter $p$ is related to the average process (entanglement) infidelity $r$ of a logic layer. For an $n$-qubit system ($d = 2^n$), the correspondence is:

$$r = \frac{d-1}{d} \cdot (1-p) = \frac{4^n-1}{4^n} \cdot (1-p)$$

as used in MRB (Proctor et al., 2021). Once $p$ is extracted by nonlinear least-squares fitting of $\overline{S}_d$ across depths, $r$ directly quantifies the average error per layer.

In the broader mirror benchmarking framework, connection to channel fidelity and unitarity is afforded via expansion in the Pauli basis:

• The twirl’s quadratic contraction yields

$S(L) = A' \, u^{L} + B,$

where $u$ denotes the unitarity,

$$u = \frac{1}{d^2-1} \operatorname{Tr} \left[ \Pi_2 \mathcal{E}^\dagger \Pi_2 \mathcal{E} \right]$$

• For purely stochastic Pauli noise, $u$ reduces to the squared average depolarization parameter; in this case, process fidelity satisfies the bounds

$$\frac{1}{d^2}\left(1+(d^2-1)u\right) \leq F(\mathcal{E}) \leq \frac{1}{d^2}\left(1+(d^2-1)\sqrt{u}\right)$$

• The degree to which $u$ deviates from the square of the “Pauli diagonal” $f$ quantifies the coherence of underlying noise (Mayer et al., 2021).

4. SPAM Robustness and Error Mitigation

A defining advantage of mirror-circuit fidelity estimation protocols is robustness to state-preparation and measurement (SPAM) errors. This robustness is achieved via:

• Random Pauli “twirling” layers between Clifford logic layers, which symmetrize coherent errors into Pauli channels, eliminating systematic error buildup in the decay rate extraction.
• Initial and final random one-qubit Cliffords implement local 2-designs, scrambling SPAM contributions and further isolating the decay dynamics to stochastic noise in the logic layers.
• The effective polarization $S(C)$ leverages a Hamming-weight sum that cancels readout biases to leading order.

Consequently, SPAM contributions enter as multiplicative prefactors (e.g., $A$ in $\overline{S}_d = Ap^d$), not entering into the decay parameter $p$. The model offset $B$ is driven to near zero (unlike conventional RB, where $B$ may reflect persistent SPAM contributions), due to the effect of Pauli twirling and post-selection (Proctor et al., 2021, Mayer et al., 2021).

5. Scalability, Resource Costs, and Experimental Demonstrations

Mirror circuit fidelity estimation protocols, and MRB specifically, have demonstrated scalability to quantum systems far exceeding the reach of conventional randomized benchmarking:

• Circuit gate cost is $O(n)$ one- and two-qubit gates per logic layer, a linear overhead that contrasts favorably with the $O(n^2 / \ln n)$ scaling required to realize generic $n$-qubit Cliffords in standard RB.
• Numerical simulations on up to 225 qubits in a $15\times 15$ lattice configuration demonstrated, for physically realistic error rates (0.1–1%), that infidelity estimates $r_\Omega$ tracked true error rates $\epsilon_\Omega$ to within $\pm 20\%$ (relative error) across 900 runs.
• MRB’s scaling permits investigation into crosstalk phenomena: in a 16-qubit IBM Q Rueschlikon experiment, MRB quantified two-qubit crosstalk by mapping $r_\Omega$ as a function of region size, showing divergence from error rates predicted by uncorrelated single- and two-qubit gate calibration (Proctor et al., 2021).

A comparison of experimental viability is summarized below:

Protocol	Max. Qubits Demonstrated	Gate Cost per Circuit	SPAM Robustness
Standard RB	$\sim$5	$O(n^2/\ln n)$	Moderate
Direct RB	$\sim$5–16	$O(n)$	Moderate
MRB / Mirror RB	16 (expt), 225 (sim)	$O(n)$	High

MRB remains feasible as $n$ increases due to its favorable signal falloff and linear gate complexity, enabling benchmarking in regimes beyond the reach of alternative techniques.

6. Quantum Query Complexity and Algorithmic Estimation

Beyond physical implementation, recent advances in quantum algorithms establish optimal query complexity bounds for estimating the fidelity (or infidelity) associated with mirror circuits in the black-box access model. When given access to circuit state-preparations and their inverses, the fidelity

$$F_{\rm mirror} = |\langle 0 | U_{\rm mirror} | 0 \rangle|$$

can be estimated to additive error $\varepsilon$ using $\Theta(1/\varepsilon)$ queries—provably optimal via a matching lower bound (Wang, 29 Aug 2024). This is realized using square-root amplitude estimation:

• Prepare the overlap-encoding unitary $R = U^\dagger V$ ($V = I$ for $F_{\rm mirror}$).
• Define a unitary $W'$ that encodes $F_{\rm mirror}$ in the amplitude of a specific basis state.
• Use generalized amplitude estimation (phase estimation on Grover-like iterations of $W'$) to estimate this amplitude within $\varepsilon$, using $\Theta(1/\varepsilon)$ queries. This framework provides fundamental lower bounds for the number of mirror circuit (and inverse) invocations required for fidelity estimation to a specified precision in fully coherent quantum-access scenarios, with the standard overheads for controlled operation construction and ancilla qubits.

7. Applications, Best Practices, and Limitations

Mirror circuit fidelity estimation serves as a versatile tool for system-level characterization of quantum processors, applicable to:

• Quantifying average logic-layer infidelity across increasing qubit counts,
• Revealing error growth mechanisms such as crosstalk,
• Estimating noise coherence properties via unitarity,
• Providing a direct metric for hardware performance benchmarking.

Key practical guidelines for effective implementation include:

• Select sequence lengths exceeding the qubit number ($L > n$) to ensure sufficient approximation of 2-design twirling and robust noise averaging.
• Incorporate randomized compiling (Pauli twirls) to mitigate coherent error contributions unless direct assessment of noise coherence is desired.
• Employ analytical determination of offset $B$ and SPAM-insensitive observables whenever possible.
• Use substantial circuit and shot sample sizes per depth, with bootstrapping methods for error estimation.

Limitations include the requirement for inversion-symmetric gate sets and sufficiently uniform layer noise for strict interpretation of decay rates, as well as manageable circuit depth for device coherence constraints. A plausible implication is that further development of hybrid protocols may be needed to fully leverage these techniques in platforms with highly non-uniform or strongly non-Markovian noise profiles.

Mirror circuit fidelity estimation, by exploiting symmetry, SPAM resistance, and scalable measurement of exponential decay, constitutes a key methodology for deep, efficient, and reliable assessment of quantum hardware performance across present and emerging device architectures (Proctor et al., 2021, Mayer et al., 2021, Wang, 29 Aug 2024).