BQP-completeness of scattering in scalar quantum field theory

Jordan, Stephen P.; Krovi, Hari; Lee, Keith S. M.; Preskill, John

doi:10.22331/q-2018-01-08-44

Cited by 83 publications

(76 citation statements)

References 22 publications

(54 reference statements)

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“…The prospect of being able to explore quantities in quantum many-body systems, including quantum gauge field theories such as quantum chromodynamics, that require exponentially-large classical computing resources, such as for dense matter or in the timeevolution of non-equilibrium systems, is truly exciting. In this work, we have built upon foundational works by Jordan, Lee and Preskill [6][7][8][9] on how to formulate scalar field theory on quantum computers to determine properties of the scalar particle and interactions, both elastic and inelastic, between particles. In an attempt to understand the magnitude of resources required for even modest quantum computations in a simple field theory, our work has focused on the digitization of scalar field theories with only a small number of qubits per spatial lattice site.…”

Section: Discussionmentioning

confidence: 99%

“…At least in the NISQ era, it is expected that such global operations will be prohibitive both in gate fidelity as well as coherence time. For this reason, the finite-difference form of the gradient operator, demanding only local interactions between the qubit registers at neighboring sites, appears to be optimal [6][7][8][9].…”

Section: +1 Dimensional λφ 4 Scalar Field Theorymentioning

confidence: 99%

“…In a series of foundational papers, Jordan, Lee and Preskill (JLP) formulated and analyzed scalar field theories for quantum computers [6][7][8][9] and estimated the resource-requirement scaling of calculations of static properties and of elastic and inelastic particle scattering processes determined through direct time evolution. A real scalar field, φ(x), is discretized on a spatial lattice using techniques that are standard in lattice QCD (LQCD) calculations using classical computers.…”

Section: Introductionmentioning

confidence: 99%

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Digitization of scalar fields for quantum computing

2019

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Qubit, operator and gate resources required for the digitization of lattice λφ 4 scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan, Lee and Preskill, with a focus towards noisy intermediate-scale quantum (NISQ) devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin, Spentzouris, Amundson and Harnik building on the work of Somma, provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan, Lee and Preskill, and eigenstates of a harmonic oscillator, to describe 0 + 1-and d + 1-dimensional scalar field theory. We show how techniques associated with improved actions, which are heavily utilized in Lattice QCD calculations to systematically reduce lattice-spacing artifacts, can be used to reduce the impact of the field digitization in λφ 4 , but are found to be inferior to a complete digitization-improvement of the Hamiltonian using a Quantum Fourier Transform. When the Nyquist-Shannon sampling theorem is satisfied, digitization errors scale as | log | log | dig ||| ∼ n Q (number of qubits describing the field at a given spatial site) for the low-lying states, leaving the familiar power-law lattice-spacing and finite-volume effects that scale as | log | latt || ∼ N Q (total number of qubits in the simulation). For localized(delocalized) field-space wavefunctions, it is found that n Q ∼ 4(7) qubits per spatial lattice site are sufficient to reduce theoretical digitization errors below error contributions associated with approximation of the time-evolution operator and noisy implementation on near-term quantum devices. 1 1 Only classical computing resources have been used to obtain the results presented in this work.

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Section: Discussionmentioning

confidence: 99%

Section: +1 Dimensional λφ 4 Scalar Field Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Digitization of scalar fields for quantum computing

2019

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“…The study of circuit complexity in quantum field theory is in its infancy; only a few cases have been studied to date and much remains unexplored. In [86][87][88], it was shown that simulation of several field theoretic observables on a quantum computer has an exponential advantage over classical algorithms which use perturbative Feynman diagrams. For our purpose, we will adhere to the notion of complexity associated with a quantum circuit-the task is to prepare the 'target state' (for us, this is the time evolved ground state of some Hamiltonian) by a quantum circuit starting from the suitable 'reference state', and make this circuit as efficient as possible.…”

Section: Introductionmentioning

confidence: 99%

Time evolution of complexity: a critique of three methods

Ali

Bhattacharyya

Haque

et al. 2019

J. High Energ. Phys.

112

132

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In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the informationtheoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour-the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.

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“…On the other hand, as Feynman proposed [1], quantum computers may be able to carry out the simulation more efficiently than classical computers. This led to a large body of research dealing with quantum algorithms for Hamiltonian simulation , with numerous applications to problems in physics and chemistry [30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Divide and conquer approach to quantum Hamiltonian simulation

Hadfield

Papageorgiou

2018

New J. Phys.

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We show a divide and conquer approach for simulating quantum mechanical systems on quantum computers. We can obtain fast simulation algorithms using Hamiltonian structure. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under very mild assumptions.

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BQP-completeness of scattering in scalar quantum field theory

Cited by 83 publications

References 22 publications

Digitization of scalar fields for quantum computing

Digitization of scalar fields for quantum computing

Time evolution of complexity: a critique of three methods

Divide and conquer approach to quantum Hamiltonian simulation

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