A Theory of Quantum Subspace Diagonalization

Epperly, Ethan N.; Lin, Lin; Nakatsukasa, Yuji

doi:10.1137/21m145954x

Cited by 24 publications

(34 citation statements)

References 26 publications

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“…We now turn to the discussion of the calculations of excitation energies, for which subspace type methods − are suitable. These methods can be viewed as a quantum analog of CI and its variants, e.g., the selected CI approach .…”

Section: Methodsmentioning

confidence: 99%

Quantum Simulations of Fermionic Hamiltonians with Efficient Encoding and Ansatz Schemes

Huang

Sheng

Govoni

et al. 2023

J. Chem. Theory Comput.

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We propose a computational protocol for quantum simulations of fermionic Hamiltonians on a quantum computer, enabling calculations on spin defect systems which were previously not feasible using conventional encodings and a unitary coupled-cluster ansatz of variational quantum eigensolvers. We combine a qubit-efficient encoding scheme mapping Slater determinants onto qubits with a modified qubit-coupled cluster ansatz and noise-mitigation techniques. Our strategy leads to a substantial improvement in the scaling of circuit gate counts and in the number of required qubits, and to a decrease in the number of required variational parameters, thus increasing the resilience to noise. We present results for spin defects of interest for quantum technologies, going beyond minimum models for the negatively charged nitrogen vacancy center in diamonds and the double vacancy in 4H silicon carbide (4H-SiC) and tackling a defect as complex as negatively charged silicon vacancy in 4H-SiC for the first time.

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Section: Methodsmentioning

confidence: 99%

Quantum Simulations of Fermionic Hamiltonians with Efficient Encoding and Ansatz Schemes

Huang

Sheng

Govoni

et al. 2023

J. Chem. Theory Comput.

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“…In the numerical study, we focus on the optimisation method and Trotter formula instead of the iterative method and LOR formula. Quantum subspace diagonalisation (QSD) or quantum Lanczos [18,[43][44][45][46] is a way to reduce the error in GSS, and we also demonstrate this method in the numerical study. Following Ref.…”

Section: Our Algorithm Effectively Prepares the State G(h)|ψ(0) And ...mentioning

confidence: 99%

“…To obtain a stable inverse in the numerical calculation, we apply a truncation on eigenvalues of A [46]. We suppose eigenvalues are λ 1 , λ 2 , .…”

Section: G Details Of the Numerical Simulationmentioning

confidence: 99%

Error-resilient Monte Carlo quantum simulation of imaginary time

Huo

Liu

2023

Quantum

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Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. The conventional algorithm quantum phase estimation uses deep circuits and requires fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit shallow circuits. In this work, we combine quantum Monte Carlo with quantum computing and propose an algorithm for simulating the imaginary-time evolution and solving the ground-state problem. By sampling the real-time evolution operator with a random evolution time according to a modified Cauchy-Lorentz distribution, we can compute the expected value of an observable in imaginary-time evolution. Our algorithm approaches the exact solution given a circuit depth increasing polylogarithmically with the desired accuracy. Compared with quantum phase estimation, the Trotter step number, i.e. the circuit depth, can be thousands of times smaller to achieve the same accuracy in the ground-state energy. We verify the resilience to Trotterisation errors caused by the finite circuit depth in the numerical simulation of various models. The results show that Monte Carlo quantum simulation is promising even without a fully fault-tolerant quantum computer.

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“…Even for real-time evolution employing a simple linear time grid, the error of our spectral approximation can be bounded based on an extension of the Kaniel-Paige-Saad formalism [32,[43][44][45]. In particular, we establish an error bound through the following theorems.…”

Section: Proof Of Multi-step Convergencementioning

confidence: 99%

Real-Time Krylov Theory for Quantum Computing Algorithms

Yizhi¹,

Klymko²,

Sud³

et al. 2022

Preprint

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Quantum computers provide alternative avenues to access ground and excited state properties of systems difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues using quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, applicable to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution methods, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.

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A Theory of Quantum Subspace Diagonalization

Cited by 24 publications

References 26 publications

Quantum Simulations of Fermionic Hamiltonians with Efficient Encoding and Ansatz Schemes

Quantum Simulations of Fermionic Hamiltonians with Efficient Encoding and Ansatz Schemes

Error-resilient Monte Carlo quantum simulation of imaginary time

Real-Time Krylov Theory for Quantum Computing Algorithms

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