We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. Our algorithms start from a classical data structure in which the matrix of interest is specified in the Pauli basis. For $N\times N$ Hermitian matrices, the space cost is $\log(N)+1$ qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size $O(N^2)$ , when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. Our algorithms start from a classical data structure in which the matrix of interest is specified in the Pauli basis. For $N\times N$ Hermitian matrices, the space cost is $\log(N)+1$ qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size $O(N^2)$, when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

arXiv:2302.01873Feb 3, 20231 citation

Randomized Adiabatic Quantum Linear Solver Algorithm with Optimal Complexity Scaling and Detailed Running Costs

David Jennings, Matteo Lostaglio, Sam Pallister +2 more

Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Subaşı . [Phys. Rev. Lett. 122, 060504 (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number κ of the linear system and the target error ϵ . Here we go beyond these results in several ways. Firstly, using filtering [Lin and Tong, Quantum 4, 361 (2020)] and Poissonization techniques [Cunningham and Roland, ArXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling O ( κ log ( 1 / ϵ ) ) —an exponential improvement in ϵ , and a shaving of a log κ scaling factor in κ . Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation—which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is 837 κ at ϵ = 10 − 10 for Hermitian matrices.Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Subaşı . [Phys. Rev. Lett. 122, 060504 (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number κ of the linear system and the target error ϵ . Here we go beyond these results in several ways. Firstly, using filtering [Lin and Tong, Quantum 4, 361 (2020)] and Poissonization techniques [Cunningham and Roland, ArXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling O ( κ log ( 1 / ϵ ) ) —an exponential improvement in ϵ , and a shaving of a log κ scaling factor in κ . Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation—which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is 837 κ at ϵ = 10 − 10 for Hermitian matrices.

arXiv:2305.11352PRX QuantumOct 7, 20256 citations

Randomized Quantum Singular Value Transformation

Xinzhao Wang, Yuxin Zhang, Hazra, Soumyabrata +3 more

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.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.

arXiv:2510.06851Oct 8, 20250 citations

Quantum sampling algorithms for quantum state preparation and matrix block-encoding

Jessica Lemieux, Matteo Lostaglio, Sam Pallister +4 more

The problems of quantum state preparation and matrix block-encoding are ubiquitous in quantum computing: they are crucial parts of various quantum algorithms for the purpose for initial state preparation as well as loading problem relevant data. We first present an algorithm based on QRS that prepares a quantum state $|ψ_f\rangle \propto \sum^N_{x=1} f(x)|x\rangle$ . When combined with efficient reference states the algorithm reduces the cost of quantum state preparation substantially, if certain criteria on $f$ are met. When the preparation of the reference state is not the dominant cost, and the function $f$ and relevant properties are efficiently computable or provided otherwise with cost $o(N)$ , the QRS-based method outperforms the generic state preparation algorithm, which has cost $O(N)$ . We demonstrate the detailed performance (in terms of the number of Toffoli gates) of the QRS-based algorithm for quantum states commonly appearing in quantum applications, e.g., those with coefficients that obey power law decay, Gaussian, and hyperbolic tangent, and compare it with other methods. Then, we adapt QRS techniques to the matrix block-encoding problem and introduce a QRS-based algorithm for block-encoding a given matrix $A = \sum_{ij} A_{ij} |i\rangle \langle j|$ . We work out rescaling factors for different access models, which encode how the information about the matrix is provided to the quantum computer. We exemplify these results for a particular Toeplitz matrix with elements $A_{\mathbf{ij}}= 1/\|{\mathbf{i}}-{\mathbf{j}}\|^2$ , which appears in quantum chemistry, and PDE applications, e.g., when the Coulomb interaction is involved. Our work unifies, and in certain ways goes beyond, various quantum state preparation and matrix block-encoding methods in the literature, and gives detailed performance analysis of important examples that appear in quantum applications.The problems of quantum state preparation and matrix block-encoding are ubiquitous in quantum computing: they are crucial parts of various quantum algorithms for the purpose for initial state preparation as well as loading problem relevant data. We first present an algorithm based on QRS that prepares a quantum state $|ψ_f\rangle \propto \sum^N_{x=1} f(x)|x\rangle$. When combined with efficient reference states the algorithm reduces the cost of quantum state preparation substantially, if certain criteria on $f$ are met. When the preparation of the reference state is not the dominant cost, and the function $f$ and relevant properties are efficiently computable or provided otherwise with cost $o(N)$, the QRS-based method outperforms the generic state preparation algorithm, which has cost $O(N)$. We demonstrate the detailed performance (in terms of the number of Toffoli gates) of the QRS-based algorithm for quantum states commonly appearing in quantum applications, e.g., those with coefficients that obey power law decay, Gaussian, and hyperbolic tangent, and compare it with other methods. Then, we adapt QRS techniques to the matrix block-encoding problem and introduce a QRS-based algorithm for block-encoding a given matrix $A = \sum_{ij} A_{ij} |i\rangle \langle j|$. We work out rescaling factors for different access models, which encode how the information about the matrix is provided to the quantum computer. We exemplify these results for a particular Toeplitz matrix with elements $A_{\mathbf{ij}}= 1/\|{\mathbf{i}}-{\mathbf{j}}\|^2$, which appears in quantum chemistry, and PDE applications, e.g., when the Coulomb interaction is involved. Our work unifies, and in certain ways goes beyond, various quantum state preparation and matrix block-encoding methods in the literature, and gives detailed performance analysis of important examples that appear in quantum applications.

arXiv:2405.11436May 19, 20240 citations

Fast and Practical Quantum-Inspired Classical Algorithms for Solving Linear Systems

Qian Zuo, Tongyang Li

We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^m$ , we propose classical algorithms that produce a data structure for the solution $x\in\mathbb{R}^{n}$ of the linear system $Ax=b$ with the ability to sample and query its entries. The resulting $x$ satisfies $\|x-A^{+}b\|\leqε\|A^{+}b\|$ , where $\|\cdot\|$ is the spectral norm and $A^+$ is the Moore-Penrose inverse of $A$ . Our algorithm has time complexity $\widetilde{O}(κ_F^4/κε^2)$ in the general case, where $κ_{F} =\|A\|_F\|A^+\|$ and $κ=\|A\|\|A^+\|$ are condition numbers. Compared to the prior state-of-the-art result [Shao and Montanaro, arXiv:2103.10309v2], our algorithm achieves a polynomial speedup in condition numbers. When $A$ is $s$ -sparse, our algorithm has complexity $\widetilde{O}(s κ\log(1/ε))$ , matching the quantum lower bound for solving linear systems in $κ$ and $1/ε$ up to poly-logarithmic factors [Harrow and Kothari]. When $A$ is $s$ -sparse and symmetric positive-definite, our algorithm has complexity $\widetilde{O}(s\sqrtκ\log(1/ε))$ . Technically, our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems, where we propose two new methods with speedups: quantum-inspired Kaczmarz method with momentum and quantum-inspired coordinate descent method with momentum. Their analysis exploits careful decomposition of the momentum transition matrix and the application of novel spectral norm concentration bounds for independent random matrices. Finally, we also conduct numerical experiments for our algorithms on both synthetic and real-world datasets, and the experimental results support our theoretical claims.We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^m$, we propose classical algorithms that produce a data structure for the solution $x\in\mathbb{R}^{n}$ of the linear system $Ax=b$ with the ability to sample and query its entries. The resulting $x$ satisfies $\|x-A^{+}b\|\leqε\|A^{+}b\|$, where $\|\cdot\|$ is the spectral norm and $A^+$ is the Moore-Penrose inverse of $A$. Our algorithm has time complexity $\widetilde{O}(κ_F^4/κε^2)$ in the general case, where $κ_{F} =\|A\|_F\|A^+\|$ and $κ=\|A\|\|A^+\|$ are condition numbers. Compared to the prior state-of-the-art result [Shao and Montanaro, arXiv:2103.10309v2], our algorithm achieves a polynomial speedup in condition numbers. When $A$ is $s$-sparse, our algorithm has complexity $\widetilde{O}(s κ\log(1/ε))$, matching the quantum lower bound for solving linear systems in $κ$ and $1/ε$ up to poly-logarithmic factors [Harrow and Kothari]. When $A$ is $s$-sparse and symmetric positive-definite, our algorithm has complexity $\widetilde{O}(s\sqrtκ\log(1/ε))$. Technically, our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems, where we propose two new methods with speedups: quantum-inspired Kaczmarz method with momentum and quantum-inspired coordinate descent method with momentum. Their analysis exploits careful decomposition of the momentum transition matrix and the application of novel spectral norm concentration bounds for independent random matrices. Finally, we also conduct numerical experiments for our algorithms on both synthetic and real-world datasets, and the experimental results support our theoretical claims.

arXiv:2307.06627Jul 13, 20230 citations

Hamiltonian Simulation and Linear Combination of Unitary Decomposition of Structured Matrices

Robin Ollive, Stéphane Louise

To process a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operators, or gates. These operators are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BEing). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hamiltonian (LCH) or a Linear Combination of nitaries (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. In this work, we propose Hamiltonian matrices used to map the problem of interest, and that behave like a single qubit when expressed in the appropriate basis. It leads to a framework, the Special Tripotent Hamiltonian (STH) Framework, able to implement most of the typical problems considered for quantum computing. These methods address many problems implemented on QPUs, ranging from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), sparse matrices, and higher-order optimization problems. This work underlines interesting properties associated with the STH basic gate decomposition. These include the ability to switch between LCH and LCU, map non-Hermitian problems, and construct the quantum circuit queries required for quantum computing. We also provide a list of STH that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.To process a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operators, or gates. These operators are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BEing). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hamiltonian (LCH) or a Linear Combination of nitaries (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. In this work, we propose Hamiltonian matrices used to map the problem of interest, and that behave like a single qubit when expressed in the appropriate basis. It leads to a framework, the Special Tripotent Hamiltonian (STH) Framework, able to implement most of the typical problems considered for quantum computing. These methods address many problems implemented on QPUs, ranging from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), sparse matrices, and higher-order optimization problems. This work underlines interesting properties associated with the STH basic gate decomposition. These include the ability to switch between LCH and LCU, map non-Hermitian problems, and construct the quantum circuit queries required for quantum computing. We also provide a list of STH that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.

quant-pharXiv:2603.17816Mar 19, 20260 citations

N\times N

Hermitian matrices, the space cost is

\log(N)+1

qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size

O(N^2)

, when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

|ψ_f\rangle \propto \sum^N_{x=1} f(x)|x\rangle

. When combined with efficient reference states the algorithm reduces the cost of quantum state preparation substantially, if certain criteria on

f

are met. When the preparation of the reference state is not the dominant cost, and the function

f

and relevant properties are efficiently computable or provided otherwise with cost

o(N)

, the QRS-based method outperforms the generic state preparation algorithm, which has cost

O(N)

. We demonstrate the detailed performance (in terms of the number of Toffoli gates) of the QRS-based algorithm for quantum states commonly appearing in quantum applications, e.g., those with coefficients that obey power law decay, Gaussian, and hyperbolic tangent, and compare it with other methods. Then, we adapt QRS techniques to the matrix block-encoding problem and introduce a QRS-based algorithm for block-encoding a given matrix

A = \sum_{ij} A_{ij} |i\rangle \langle j|

. We work out rescaling factors for different access models, which encode how the information about the matrix is provided to the quantum computer. We exemplify these results for a particular Toeplitz matrix with elements

A_{\mathbf{ij}}= 1/\|{\mathbf{i}}-{\mathbf{j}}\|^2

, which appears in quantum chemistry, and PDE applications, e.g., when the Coulomb interaction is involved. Our work unifies, and in certain ways goes beyond, various quantum state preparation and matrix block-encoding methods in the literature, and gives detailed performance analysis of important examples that appear in quantum applications.

We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix

A\in\mathbb{R}^{m\times n}

and a vector

b\in\mathbb{R}^m

, we propose classical algorithms that produce a data structure for the solution

x\in\mathbb{R}^{n}

of the linear system

Ax=b

with the ability to sample and query its entries. The resulting

x

satisfies

\|x-A^{+}b\|\leqε\|A^{+}b\|

, where

\|\cdot\|

is the spectral norm and

A^+

is the Moore-Penrose inverse of

A

. Our algorithm has time complexity

\widetilde{O}(κ_F^4/κε^2)

in the general case, where

κ_{F} =\|A\|_F\|A^+\|

and

κ=\|A\|\|A^+\|

are condition numbers. Compared to the prior state-of-the-art result [Shao and Montanaro, arXiv:2103.10309v2], our algorithm achieves a polynomial speedup in condition numbers. When

A

s

-sparse, our algorithm has complexity

\widetilde{O}(s κ\log(1/ε))

, matching the quantum lower bound for solving linear systems in

κ

and

1/ε

up to poly-logarithmic factors [Harrow and Kothari]. When

A

s

-sparse and symmetric positive-definite, our algorithm has complexity

\widetilde{O}(s\sqrtκ\log(1/ε))

. Technically, our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems, where we propose two new methods with speedups: quantum-inspired Kaczmarz method with momentum and quantum-inspired coordinate descent method with momentum. Their analysis exploits careful decomposition of the momentum transition matrix and the application of novel spectral norm concentration bounds for independent random matrices. Finally, we also conduct numerical experiments for our algorithms on both synthetic and real-world datasets, and the experimental results support our theoretical claims.