Shadow Quantum Singular Value Transformation with Shallow Quantum Circuits
Abstract
We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state |ψ⟩, a Hermitian matrix H, a polynomial f, and a set of observables {O1,…,Om}, the goal is to estimate ⟨ψ|f(H)†Oj f(H)|ψ⟩ for all j∈{1,…,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⟩ 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⟩ without prior knowledge of the observables or explicit preparation of the target state proportional to f(H)|ψ⟩. 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.
Related
- Polynomial-time additive-error estimation of output probabilities for shallow quantum circuits
- Classical Pre-Training of Large Shallow Parametrized Quantum Circuits through Pauli Propagation and deployment on Quantum Hardware
- Experimentally Testable Quantum Advantage in Shallow Circuits
- Distilling Datasets into Shallow Circuits for Quantum Machine Learning
- Parallel classical simulation of noisy shallow circuits: no quantum advantage in 1D
BibTeX
@misc{chia2026shadow,
title = {{Shadow Quantum Singular Value Transformation with Shallow Quantum Circuits}},
author = {Nai-Hui Chia and Hyunseong Kim and Chia-Ying Lin and Yu-Ching Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.40167},
eprint = {2609.40167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40167},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Shadow Quantum Singular Value Transformation with Shallow Quantum Circuits
AU - Nai-Hui Chia
AU - Hyunseong Kim
AU - Chia-Ying Lin
AU - Yu-Ching Shen
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.40167
UR - https://arxiv.org/abs/2609.40167
AB - We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state |ψ⟩, a Hermitian matrix H, a polynomial f, and a set of observables {O_1,…,O_m}, the goal is to estimate ⟨ψ|f(H)^†O_j f(H)|ψ⟩ for all j∈{1,…,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⟩ 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⟩ without prior knowledge of the observables or explicit preparation of the target state proportional to f(H)|ψ⟩. 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.
ER -