Pauli-resolved virtual distillation

Abstract

Learning the full Pauli profile of the virtually distilled quantum state $\rho^m/\text{tr}(\rho^m)$ has so far required exponentially many copies of $\rho$. We show that all $4^n$ squared Pauli moments $[\text{tr}(P\rho^m)]^2$ can be learned to additive error $\varepsilon$ with confidence $1-\delta$ from one $2m$-replica measurement setting using $O(m[n+\log(1/\delta)]/\varepsilon^2)$ copies. This is an exponential speedup in the system size $n$ over previous protocols. For $m = 2$, we propose the coherent Bell difference sampling circuit that realizes this measurement on current devices. The speedup originates from paired replicas that cancel the anticommutation signs of Pauli operators, collapsing the incompatibility of the Pauli family. We certify this collapse by introducing the quantum Bernstein norm, a computable incompatibility measure for nonlinear functionals. A further information-theoretic $m$-replica protocol recovers the signed moments $\text{tr}(P\rho^m)$ using $O(m[n+\log(1/\delta)]/\varepsilon^4)$ copies. For fixed $m$ in the stated lower-bound regime, it attains the optimal replica number, since any protocol with fewer replicas requires exponentially many copies.