Randomized Quantum Singular Value Transformation

TL;DR

This work introduces the first randomized algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework for many quantum algorithms, and establishes a fundamental lower bound showing that the quadratic dependence on the polynomial degree is optimal within this framework.

Abstract

We introduce the first randomized algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework for many quantum algorithms. Standard implementations of QSVT rely on block encodings of the Hamiltonian, which are costly to construct, requiring a logarithmic number of ancilla qubits, intricate multi-qubit control, and circuit depth scaling linearly with the number of Hamiltonian terms. In contrast, our algorithms use only a single ancilla qubit and entirely avoid block encodings. We develop two methods: (i) a direct randomization of QSVT, where block encodings are replaced by importance sampling, and (ii) an approach that integrates qDRIFT into the generalized quantum signal processing framework, with the dependence on precision exponentially improved through classical extrapolation. Both algorithms achieve gate complexity independent of the number of Hamiltonian terms, a hallmark of randomized methods, while incurring only quadratic dependence on the degree of the target polynomial. We identify natural parameter regimes where our methods outperform even standard QSVT, making them promising for early fault-tolerant quantum devices. We also establish a fundamental lower bound showing that the quadratic dependence on the polynomial degree is optimal within this framework. We apply our framework to two fundamental tasks: solving quantum linear systems and estimating ground-state properties of Hamiltonians, obtaining polynomial advantages over prior randomized algorithms. Finally, we benchmark our ground-state property estimation algorithm on electronic structure Hamiltonians and the transverse-field Ising model with long-range interactions. In both cases, our approach outperforms prior work by several orders of magnitude in circuit depth, establishing randomized QSVT as a practical and resource-efficient alternative for early fault-tolerant quantum devices.

Figures (4)

• Figure 1: Comparison of the circuit depth of the different quantum algorithms for ground state property estimation when applied to the electronic structure Hamiltonians of propane (in STO-3G basis), carbon dioxide (in 6-31G basis), and ethane (in 6-31G basis). We assume that the initial "guess" state has an overlap of $\gamma=0.1$ with the ground state of the Hamiltonian. We plot the circuit depth per coherent run as a function of the error tolerance $\varepsilon$. The circuit depth of randomized QSVT is significantly shorter than the other approaches considered here, for all three molecules.
• Figure 2: Comparison of the overall gate complexity (total number of single and two-qubit gates needed) of our method with other randomized methods chakraborty2024implementingwang2024qubit for ground state property estimation when applied to the electronic structure Hamiltonians of propane (in STO-3G basis), carbon dioxide (in 6-31G basis), and ethane (in 6-31G basis). We fix the error tolerance $\varepsilon=0.01$, and vary $\gamma$, the overlap of the initial "guess" state with the ground state of the Hamiltonian. Our method requires substantially fewer gates than prior randomized methods for all three molecules.
• Figure 3: Comparison of the circuit depth of different quantum algorithms for ground state property estimation as a function of the number of qubits $n$ for the two Transverse Field Ising Model Hamiltonians considered in this work. (To the left) Transverse-field Ising model with long-range interactions, with the Hamiltonian parameters set to $h=3$, $J=1$, $\alpha=3$, $\varepsilon = 0.01$, $\gamma=0.1$ (See Eq. \ref{['eq:Ham-TFIM-long-range']}). The spectral gap is fixed to $\Delta=3$. (To the right) Transverse-field Ising model with hybrid interactions, with the Hamiltonian parameters $h=3$, $J=1$, $g=0.1$, $\alpha=3$, $\varepsilon = 0.01$, $\gamma=0.1$ (See Eq. \ref{['eq:Ham-TFIM-hybrid']}). In both cases, the circuit depth of ground state property estimation by randomized QSVT is shorter as compared to the other methods.
• Figure 4: Spectral gap of the one-dimensional long-range transverse-field Ising model with $J=1$, $h=3$, and $\alpha=3$, as a function of the number spins $n$.

Theorems & Definitions (52)

• Lemma 2.1: QSP, Corollary 8 of gilyen2019quantum
• Definition 2.2: Block encoding
• Definition 2.3: Singular value transformation
• Definition 2.4: Definition 8 of gilyen2019quantum
• Lemma 2.5: QSVT, Theorem 10 of gilyen2019quantum
• Theorem 2.6: Combining Corollary 5 and Theorem 6 of motlagh2024generalized
• Lemma 2.7: dollard1984product
• Lemma 2.8: Richardson extrapolation, see watson2024randomlylow2019well
• Lemma 2.9
• proof