Destructive Error Interference in Product-Formula Lattice Simulation

Tran, Minh C.; Chu, Su-Kuan; Su, Yuan; Childs, Andrew M.; Gorshkov, Alexey V.

doi:10.1103/physrevlett.124.220502

Cited by 35 publications

(34 citation statements)

References 13 publications

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“…In this section, we discuss how to generate the time evolution of a 1d fermion chain with nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping terms using only twosite gates acting on neighboring sites on a fermionic digital quantum simulator. 4 Consider a 1d fermion chain with one complex fermion on each site j. The fermion operators c j , c † j satisfy the canonical anticommutation relations…”

Section: B 1d Fermion Chain With Next-nearest-neighbor Hopping Termsmentioning

confidence: 99%

“…Product formulas approximate a desired unitary operator with a product of simpler operator exponentials. As a practical tool, product formulas are widely used in quantum simulation of condensed matter models and quantum chemistry problems [1][2][3][4][5][6][7][8][9], as well as quantum Monte Carlo and statistical physics problems [10][11][12][13]. The simplest example is the first-order Lie-Trotter product formula…”

mentioning

confidence: 99%

“…c 5 + ⋯ + h.c. 4. The following derivation uses fermionic operators c, c † , which can be transformed to Pauli operators by the Jordan-Wigner transformation.…”

mentioning

confidence: 99%

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Efficient Product Formulas for Commutators and Applications to Quantum Simulation

Chen,

Childs,

Hafezi

et al. 2021

Preprint

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We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the thirdorder formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest-and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases.

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Section: B 1d Fermion Chain With Next-nearest-neighbor Hopping Termsmentioning

confidence: 99%

mentioning

confidence: 99%

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Efficient Product Formulas for Commutators and Applications to Quantum Simulation

Chen,

Childs,

Hafezi

et al. 2021

Preprint

Self Cite

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show abstract

“…We refer to Refs. [14,17,19,21,31,34] for error analyses of product formula methods for more general problems. We note that due to the simplicity of the product formula method, it is widely used in experimental implementations of Hamiltonian simulation with near-term quantum devices [4,11,12].…”

Section: Introductionmentioning

confidence: 99%

Exploiting anticommutation in Hamiltonian simulation

Zhao

Yuan

2021

Quantum

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Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in Hamiltonian simulation. We find that, contrary to our intuition, anti-commutative relations could also reduce the hardness of Hamiltonian simulation. Specifically, Hamiltonians with mutually anti-commutative terms are easy to simulate, as what happens with ones consisting of mutually commutative terms. Such a property is further utilized to reduce the algorithmic error or the gate complexity in the truncated Taylor series quantum algorithm for general problems. Moreover, we propose two modified linear combinations of unitaries methods tailored for Hamiltonians with different degrees of anti-commutation. We numerically verify that the proposed methods exploiting anti-commutative relations could significantly improve the simulation accuracy of electronic Hamiltonians. Our work sheds light on the roles of commutative and anti-commutative relations in simulating quantum systems.

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“…In (i) and (ii), measurements on input/output qubits are not counted. In general, the M needed will depend on the model, model parameters, tolerances, and scales as [40]:…”

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confidence: 99%

Measurement-Based Time Evolution for Quantum Simulation of Fermionic Systems

Lee,

Qin,

Raussendorf

et al. 2021

Preprint

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Quantum simulation using time evolution in phase estimation-based quantum algorithms can yield unbiased solutions of classically intractable models. But long runtimes open such algorithms to decoherence. We show how measurement-based quantum simulation uses effective time evolution via measurement to allow runtime advantages over conventional circuit-based algorithms that use real-time evolution with quantum gates. We construct a hybrid algorithm to find energy eigenvalues in fermionic models using only measurements on graph states. We apply the algorithm to the Kitaev and Hubbard chains. Resource estimates show a runtime advantage if measurements can be performed faster than gates. Our work sets the stage to allow advances in measurement precision to improve quantum simulation.

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Destructive Error Interference in Product-Formula Lattice Simulation

Cited by 35 publications

References 13 publications

Efficient Product Formulas for Commutators and Applications to Quantum Simulation

Efficient Product Formulas for Commutators and Applications to Quantum Simulation

Exploiting anticommutation in Hamiltonian simulation

Measurement-Based Time Evolution for Quantum Simulation of Fermionic Systems

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