Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

TL;DR

The paper develops a unifying framework, singular value transformation (SVT), that generalizes qubitization and quantum signal processing to apply polynomial transformations to the singular values of a block of a unitary. It provides constructive, low-cost quantum circuits for implementing P^{(SV)}(A) for any bounded odd (and even) polynomial P, and extends to robust, non-Hermitian cases via invariant subspace decompositions. The results yield broad algorithmic consequences, including direct Moore-Penrose pseudoinverse implementation, singular value thresholding and discrimination, non-commutative measurements, and exponential improvements for fractional queries, plus a powerful matrix-arithmetic toolkit based on block-encodings. It also demonstrates practical quantum machine learning applications, notably principal component regression, and develops a comprehensive theory for smooth function evaluation of Hermitian matrices with tight complexity bounds. Overall, SVT provides a versatile, scalable approach to quantum matrix algebra with far-reaching implications for quantum simulation, linear systems, walk-based algorithms, and quantum ML.

Abstract

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

Figures (1)

• Figure 1: Gates and gate sequences used for singular value transformation in Theorem \ref{['thm:singValTransformation']}. Figure \ref{['fig:CPiNot']} shows how to implement a C$_\Pi$NOT gate, and Figure \ref{['fig:Piphi']} shows how to implement $e^{i\phi(2\Pi-I)}$ using a single ancilla qubit, two C$_\Pi$NOT gates and an $e^{-i\phi\sigma_z}$ gate. Figure \ref{['fig:cPiphi']} demonstrates how to implement a controlled version of the gate $e^{i\phi^{(c)}(2\Pi-I)}$, by only controlling the single qubit gate $e^{-i\phi^{(c)}\sigma_z}$. Finally, Figure \ref{['fig:UPhi']} summarizes the complete circuit used in Theorem \ref{['thm:singValTransformation']}.

Theorems & Definitions (122)

• Definition 1: Singular value projectors
• Definition 2: C$_\Pi$NOT gate
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Lemma 6
• proof