Simulation of noisy quantum circuits using frame representations

TL;DR

The paper addresses the boundary between classical and quantum computation by introducing a frame-based framework for simulating noisy quantum circuits. It unifies Schrödinger- and Heisenberg-picture methods through quasi-probability representations, with the classical cost governed by the one-norm $||\lambda||_1$ of frame decompositions. By instantiating stabilizer, dyadic stabilizer, product-state, and Pauli-based frames, it derives inverse-threshold results (e.g., around $p_{cl}\approx 0.11$ for Clifford+$T$ with Pauli noise and $p_{cl}\approx 0.07$ in the Pauli frame) and shows how optimized decompositions via linear programs or second-order cone programs improve simulation efficiency, while no-go theorems based on magic distillation bound the ultimate limits. The framework also proposes novel frames (notably the extended Pauli frame with $a\approx 0.84$) that further reduce simulation cost in certain noise regimes. Overall, the work provides a rigorous, worst-case methodology for assessing classical simulability, offers practical algorithms for specific gate-sets, and highlights directions for developing frame-based methods to probe the quantum-classical boundary and benchmark near-term devices.

Abstract

One of the core research questions in the theory of quantum computing is to find out to what precise extent the classical simulation of a noisy quantum circuits is possible and where potential quantum advantages can set in. In this work, we introduce a unified framework for the classical simulation of quantum circuits based on frame theory, encompassing and generalizing a broad class of existing simulation strategies. Within this framework, the computational cost of a simulation algorithm is determined by the one-norm of an associated quasi-probability distribution, providing a common quantitative measure across different simulation approaches. This enables a comprehensive perspective on common methods for the simulation of noisy circuits based on different quantum resources, such as entanglement or non-stabilizerness. It further provides a clear scheme for generating novel classical simulation algorithms. Indeed, by exploring different choices of frames within this formalism and resorting to tools of convex optimization, we are able not only to obtain new insights and improved bounds for existing methods -- such as stabilizer state simulation or Pauli back-propagation -- but also to discover a new approach with an improved performance based on a generalization of the Pauli frame. We, thereby, show that classical simulation techniques can directly benefit from a perspective -- that of frames -- that goes beyond the traditional classification of quantum resources.

Figures (5)

• Figure 1: A schematic representation of a simulation algorithm based on the frame theory framework (represented here in the Schrödinger picture for illustration purposes). A path of frame elements is generated by sampling at each step a new element, based on the frame decomposition of the circuit gate acting on the previous element. At the final step, an estimator $E$ is evaluated, whose range -- and thus its sample complexity -- depends on the quantity $||\lambda||_1{}^{m+1}$.
• Figure 2: Stabilizer frames.$||\lambda^{(j)}||_1^{\,2}$ for optimal decompositions in the diagonal and dyadic stabilizer frames of the noisy $T$-gate at different depolarizing and dephasing strengths $p$. The total algorithm runtime will scale as $O(||\lambda||_1^{\,2t})$, where $t$ is the number of noisy $T$ gates in the circuit.
• Figure 3: Product frame.$||\lambda^{(j)}||_1^{\,2}$ for optimal decompositions with respect to a discrete approximation of the continuous product-state frame, plotted as a function of the depolarizing strength $p$ for the three gates in the gate-set ($H$, $T$, and $CNOT$). The frame is constructed from local factors selected from a finite set of $30$ uniformly random single-qubit states.
• Figure 4: Pauli frame.$||\lambda^{(j)}||_1^{\,2}$ for decompositions in the Pauli frame of the noisy $T$-gate at different depolarizing, dephasing and amplitude damping (A.D.) strengths. Note that the depolarizing and dephasing strengths $p$ and the amplitude damping strength $q$ are plotted in separate axes as they take values in different ranges.
• Figure 5: Extended Pauli frame.$||\lambda^{(j)}||_1^{\,2}$ for optimal decompositions in the extended Pauli frame at different depolarizing strengths $p$ (blue lines). We assume a universal gate set composed of noisy Hadamard ($H$), controlled-NOT ($CNOT$) and magic ($T$) gates. The extended Pauli frame is defined with the choice $a=0.84$. For comparison we also plot the value of $||\lambda^{(T,{\rm depol.})}||_1^{\,2}$ for the simple Pauli frame (green line).