Shadow Quantum Singular Value Transformation with Shallow Quantum Circuits

2026-09-30

AI Summary

Standard Quantum Singular Value Transformation (QSVT) relies on fully unitary block encodings, resulting in substantial circuit depth and high resource overhead. This work proposes the Shadow QSVT framework, which exploits the structure of input states and observables to estimate target expectation values directly, thereby significantly reducing resource requirements. Specifically, this study introduces three novel QSVT algorithms—state-aware, observable-aware, and classical shadow-based—and systematically optimizes them by integrating Krylov subspace approximations, historical state reuse, and classical shadow representations. Experimental evaluations demonstrate that the proposed approach substantially reduces both circuit depth and gate complexity across diverse scenarios, effectively enhancing the efficiency of quantum simulations.

Abstract

We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state $|ψ\rangle$, a Hermitian matrix $H$, a polynomial $f$, and a set of observables $\{O_1,\dots,O_m\}$, the goal is to estimate $\langleψ|f(H)^{\dagger}O_j f(H)|ψ\rangle$ for all $j\in\{1,\dots,m\}$. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of $f(H)$. It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary. We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with $H$ and $|ψ\rangle$ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from $H$, $f$, and $|ψ\rangle$ without prior knowledge of the observables or explicit preparation of the target state proportional to $f(H)|ψ\rangle$. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.

Problem

• Quantum Singular Value Transformation
• circuit depth
• observable estimation
• quantum resources

Innovation

• Shadow QSVT
• Krylov subspace
• shallow quantum circuits
• classical shadow
• observable-aware