Error Mitigation and Quantum-Assisted Simulation in the Error Corrected Regime

A standard approach to quantum computing is based on the idea of promoting a classically simulable and fault-tolerant set of operations to a universal set by the addition of “magic” quantum states. In this context, we develop a general framework to discuss the value of the available, nonideal magic resources, relative to those ideally required. We single out a quantity, the quantum-assisted robustness of magic (QROM), which measures the overhead of simulating the ideal resource with the nonideal ones through quasiprobability-based methods. This extends error mitigation techniques, originally developed for noisy intermediate-scale quantum devices, to the case where qubits are logically encoded. The QROM shows how the addition of noisy magic resources allows one to boost classical quasiprobability simulations of a quantum circuit and enables the construction of explicit protocols, interpolating between classical simulation and an ideal quantum computer.

Figure 1

Relative robustness for a single qubit in the Bloch sphere. The black dot represents the pure magic state $|T⟩$. The orange, full octahedron represents the convex hull of stabilizer states, while the orange shaded region is the set ${\mathcal{Q}}_1({\tau }_{\delta })$ representing all states achievable from the available resource [a noisy $T$ state ${\tau }_{\delta }=(1-\delta )|T⟩⟨T|+\delta I/2$ with $\delta =0.1$] by stabilizer operations. Geometrically, the relative robustness is the minimum mixing needed to bring the $T$ state inside ${\mathcal{Q}}_1({\tau }_{\delta })$ (in this case, mixing with $Z{\tau }_{\delta }Z$).

Figure 2

QROM as a function of noise. $k$th root of the QROM of ${\tau }^{\otimes k}$ with respect to ${\tau }_{\delta }^{\otimes k}$, as a function of the noise level $\delta$. $\overline{\mathcal{R}}({\tau }^{\otimes k}|{\tau }_{\delta }^{\otimes k})$ are upper bounds for the corresponding QROM, based on analytical decompositions. At $\delta ={\delta }_{\mathrm{th}}$, the $k=2$ ($k=3$) upper bound recovers exactly (approximately) the ROM $\mathcal{R}({\tau }^{\otimes k})$ of Ref. [17]. The black solid line represents the bound ${\mathcal{R}}_{\text{bound}}({\tau }^{\otimes t}|{\tau }_{\delta }^{\otimes t}{)}^{1/t}$, which for small $\delta$ goes as $1+\delta /2$. While $k=2$, 3 outperform $k=1$, the improvement is modest for small level of noise.

Figure 3

Left: error mitigation and distillation. Maximum number of $T$ gates ($t$) and Cliffords ($n_c$) that can be error mitigated with moderate overhead ($\le 100$) after 0, 1, and 2 rounds of ideal Bravyi-Haah $14\mapsto 2$ distillation (from dark to lighter). The initial noise parameters are ${\delta }_c={10}^{-5}$ and $\delta =5\times {10}^{-2}$. We took the worst-case scenario where $n_c$ are all 2-qubit Cliffords. Right: classical vs quantum-assisted simulation. Ratio between the quantum-assisted and classical number of samples $M/M_{\text{classical}}$ needed to estimate the average value of a Pauli observable with given precision, as a function of the noise $\delta$ and the $T$ count $t$ of the circuit, for $t/r=3$ (inject a third of the required $T$ states).