Signal-to-noise problems in fault-tolerant quantum computation

Abstract

Quantum computation offers a promising alternative to classical Monte Carlo methods by naturally avoiding sign problems through unitary evolution. A common misconception follows from this: because quantum simulation sidesteps the sign problem, it is often assumed to be an essentially perfect substitute for classical sampling. This is false---quantum measurements are intrinsically stochastic, and estimating observables often requires a number of samples that grows exponentially with system size, time separation, or inverse temperature, exactly as in the classical signal-to-noise problem. In this work, we analyze the origin and structure of signal-to-noise problems in quantum computation, drawing parallels with classical Monte Carlo, and demonstrate that such noise persists even in sign-problem-free regimes. We discuss strategies to mitigate it, in particular a quantum generalization of Stein control variates construction, in which unbiased estimators are built from operator commutators $[X,A]$ and their discrete diagonal counterparts, and validate the approach with exactly solvable numerical examples that quantify the achievable variance reduction. Our results underscore that variance reduction, not merely sign-problem avoidance, is required to make quantum algorithms practically viable for many-body physics.