Algorithm Zoo · Primitives
Quantum Singular Value Transformation (QSVT)
First described: Gilyén, Su, Low, Wiebe, 2018
The problem
Apply a polynomial f to the singular values of a block-encoded matrix A.
Given a block encoding of A, QSVT applies any bounded polynomial f to the singular values: U_QSVT(A) = block-encodes f(A). Unifies HHL (f(x) ≈ 1/x), amplitude estimation (f = sign/threshold), Hamiltonian simulation (f(x) = e^{-ixt}), Grover, and ground-state preparation.
Best classical
n/a (primitive)
Quantum complexity
O(d) queries to the block encoding for a degree-d polynomial
Our verdict
The conceptual unifier of quantum algorithms. Read Martyn et al. 'Grand Unification' (2021) once; it reorganizes everything you know. In practice the constants matter — d can be in the hundreds for high precision — but the framework's clarity wins.
When to use it
- Any time you need f(A) for matrix A and a known polynomial f.
- Designing new algorithms — most asymptotic improvements since 2019 are QSVT applications.
When not to use it
- When you only need a single low-degree power of A (just multiply).
- When the block encoding is the cost bottleneck — QSVT is cheap, BE is expensive.
Classical baseline
n/a
Hardware cost
1 ancilla qubit on top of the block encoding's ancillas, depth O(d · BE_depth). Phase factors (the polynomial coefficients) are computed classically via optimization — well-studied and tractable.
Key papers
- Quantum Singular Value Transformation and Beyond ↗
Gilyén, Su, Low, Wiebe · 2019 · STOC
- Grand Unification of Quantum Algorithms ↗
Martyn, Rossi, Tan, Chuang · 2021 · PRX Quantum
Deep-dive tutorials
Last verified: 2026-05-24