Optimal scaling quantum linear systems solver via discrete adiabatic theorem

Costa, Pedro C. S.; An, Dong; Sanders, Yuval R.; Yuan, Shuai; Babbush, Ryan; Berry, Dominic W.

doi:10.48550/arxiv.2111.08152

Cited by 10 publications

(30 citation statements)

References 21 publications

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“…The QSVT framework also provides algorithms for a variety of other linear algebraic routines such as implementing singular-value-threshold projectors [138,188] and matrix-vector multiplication [73]. Alternatively to QLS based on the QSVT, there is a discrete-time adiabatic (Section 3.2.2) approach [117] to the QLSP by Costa et al [99] that has optimal [160] query complexity: linear in κ and logarithmic in 1/ . Since QLS algorithms invert Hermitian matrices, they can also be used to compute the Moore-Penrose pseudoinverse of an arbitrary matrix.…”

Section: Quantum Linear Systemsmentioning

confidence: 99%

See 1 more Smart Citation

A Survey of Quantum Computing for Finance

Herman¹,

Googin²,

Liu³

et al. 2022

Preprint

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Quantum computers are expected to surpass the computational capabilities of classical computers during this decade and have transformative impact on numerous industry sectors, particularly finance. In fact, finance is estimated to be the first industry sector to benefit from quantum computing, not only in the medium and long terms, but even in the short term. This survey paper presents a comprehensive summary of the state of the art of quantum computing for financial applications, with particular emphasis on Monte Carlo integration, optimization, and machine learning, showing how these solutions, adapted to work on a quantum computer, can help solve more efficiently and accurately problems such as derivative pricing, risk analysis, portfolio optimization, natural language processing, and fraud detection. We also discuss the feasibility of these algorithms on near-term quantum computers with various hardware implementations and demonstrate how they relate to a wide range of use cases in finance. We hope this article will not only serve as a reference for academic researchers and industry practitioners but also inspire new ideas for future research.

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Section: Quantum Linear Systemsmentioning

confidence: 99%

“…These potentially reduce the original HHL's quadratic dependence on sparsity. HHL's query complexity, which is independent of the complexity of Hamiltonian simulation, is O(κ 2 / ) [99].…”

Section: Quantum Linear Systemsmentioning

confidence: 99%

A Survey of Quantum Computing for Finance

Herman¹,

Googin²,

Liu³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Such state preparation methods are efficient, provided that the rate of change of the Hamiltonian is small compared to an appropriate power of the eigenvalue gap [40]. These approaches are widely used in quantum simulation, linear systems and other algorithms [41,42,43,44]. The results of [43,44] show how to implement a discrete version of adiabatic state preparation that does not reduce down to a smooth transition of the eigenstate at the initial time to the eigenstate at the end time.…”

Section: Adiabatic State Preparationmentioning

confidence: 99%

“…These approaches are widely used in quantum simulation, linear systems and other algorithms [41,42,43,44]. The results of [43,44] show how to implement a discrete version of adiabatic state preparation that does not reduce down to a smooth transition of the eigenstate at the initial time to the eigenstate at the end time. This lack of explicit time dependence allows qubitization to be used; whereas prior to this work it was unclear how one could use the technique to perform adiabatic preparation of eigenstates.…”

Section: Adiabatic State Preparationmentioning

confidence: 99%

Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions

Watkins¹,

Wiebe²,

Roggero³

et al. 2022

Preprint

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In this work we provide a new approach for approximating an ordered operator exponential using an ordinary operator exponential that acts on the Hilbert space of the simulation as well as a finitedimensional clock register. This approach allows us to translate results for simulating time-independent systems to the time-dependent case. Our result solves two open problems in simulation. It first provides a rigorous way of using discrete time-displacement operators to generate time-dependent product and multiproduct formulas. Second, it provides a way to allow qubitization to be employed for certain time-dependent formulas. We show that as the number of qubits used for the clock grows, the error in the ordered operator exponential vanishes, as well as the entanglement between the clock register and the register of the state being acted upon. As an application, we provide a new family of multiproduct formulas (MPFs) for time-dependent Hamiltonians that yield both commutator scaling and poly-logarithmic error scaling. This in turn, means that this method outperforms existing methods for simulating physically-local, time-dependent Hamiltonians. Finally, we apply the formalism to show how qubitization can be generalized to the time-dependent case and show that such a translation can be practical if the second derivative of the Hamiltonian is sufficiently small.

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“…As with classical algorithms, matching the problem to the algorithm is crucial. The first class was pioneered by the Harrow-Hasidim-Lloyd (HHL) algorithm [12], with later improvements [3,8,20,9]. These algorithms are suitable for fault tolerant devices as they rely on subroutines such as Hamiltonian simulation for potentially long times.…”

Section: Introductionmentioning

confidence: 99%

A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity

O’Malley¹,

Subaşı²,

Golden³

et al. 2022

Preprint

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Quantum algorithms for solving linear systems of equations have generated excitement because of the potential speed-ups involved and the importance of solving linear equations in many applications. However, applying these algorithms can be challenging. The Harrow-Hassidim-Lloyd algorithm and improvements thereof require complex subroutines suitable for fault-tolerant hardware such as Hamiltonian simulation, making it ill-suited to current hardware. Variational algorithms, on the other hand, involve expensive optimization loops, which can be prone to barren plateaus and local optima. We describe a quantum algorithm for solving linear systems of equations that avoids these problems. Our algorithm is based on the Woodbury identity, which analytically describes the inverse of a matrix that is a low-rank modification of another (easily-invertible) matrix. This approach only utilizes basic quantum subroutines like the Hadamard test or the swap test, so it is well-suited to current hardware. There is no optimization loop, so barren plateaus and local optima do not present a problem. The low-rank aspect of the identity enables us to efficiently transfer information to and from the quantum computer. This approach can produce accurate results on current hardware. As evidence of this, we estimate an inner product involving the solution of a system of more than 16 million equations with 2% error using IBM's Auckland quantum computer. To our knowledge, no system of equations this large has previously been solved to this level of accuracy on a quantum computer.

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Optimal scaling quantum linear systems solver via discrete adiabatic theorem

Cited by 10 publications

References 21 publications

A Survey of Quantum Computing for Finance

A Survey of Quantum Computing for Finance

Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions

A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity

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