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arXiv:1808.10378 · Physical Review A

Digitization of scalar fields for quantum computing

Natalie Klco, Martin J. Savage A

Published May 23, 2019• 165 citations
arXivHTMLPDF

Abstract

Qubit, operator, and gate resources required for the digitization of lattice $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan et al. [Quantum Inf. Comput. 14, 1014 (2014); Science 336, 1130 (2012)], with a focus towards noisy intermediate-scale quantum devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin et al. [Phys. Rev. A 98, 042312 (2018)] building on the work of Somma [Quantum Inf. Comput. 16, 1125 (2016)], provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan et al., 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 $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$, 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 $|log|log|{\ensuremath{\epsilon}}_{\mathrm{dig}}|||\ensuremath{\sim}{n}_{Q}$ (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 $|log|{\ensuremath{\epsilon}}_{\mathrm{latt}}||\ensuremath{\sim}{N}_{Q}$ (total number of qubits in the simulation). For localized (delocalized) field-space wave functions, it is found that ${n}_{Q}\ensuremath{\sim}4(7)$ 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. Only classical computing resources have been used to obtain the results presented in this work.

DOI
10.1103/physreva.99.052335

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Qubit, operator, and gate resources required for the digitization of lattice \ensuremathλ\ensuremathϕ4\ensuremath{\lambda}{\ensuremath{\phi}}^{4}\ensuremathλ\ensuremathϕ4 scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan et al. [Quantum Inf. Comput. 14, 1014 (2014); Science 336, 1130 (2012)], with a focus towards noisy intermediate-scale quantum devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin et al. [Phys. Rev. A 98, 042312 (2018)] building on the work of Somma [Quantum Inf. Comput. 16, 1125 (2016)], provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan et al., and eigenstates of a harmonic oscillator, to describe (0+10+10+1)- and (d+1d+1d+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 \ensuremathλ\ensuremathϕ4\ensuremath{\lambda}{\ensuremath{\phi}}^{4}\ensuremathλ\ensuremathϕ4, 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∣\ensuremathϵdig∣∣∣\ensuremath∼nQ|log|log|{\ensuremath{\epsilon}}_{\mathrm{dig}}|||\ensuremath{\sim}{n}_{Q}∣log∣log∣\ensuremathϵdig​∣∣∣\ensuremath∼nQ​ (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∣\ensuremathϵlatt∣∣\ensuremath∼NQ|log|{\ensuremath{\epsilon}}_{\mathrm{latt}}||\ensuremath{\sim}{N}_{Q}∣log∣\ensuremathϵlatt​∣∣\ensuremath∼NQ​ (total number of qubits in the simulation). For localized (delocalized) field-space wave functions, it is found that nQ\ensuremath∼4(7){n}_{Q}\ensuremath{\sim}4(7)nQ​\ensuremath∼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. Only classical computing resources have been used to obtain the results presented in this work.
Qubit, operator and gate resources required for the digitization of lattice λϕ4\lambda\phi^4λϕ4 scalar field theories onto quantum computers in the NISQ era are considered, building upon the foundational work by Jordan, Lee and Preskill. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin, Spetzouris, 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+10+10+1- and 1+11+11+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 λϕ4\lambda\phi^4λϕ4, 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⁡∣ϵdig∣∣∣∼nQ|\log|\log |\epsilon_{\rm dig}|||\sim n_Q∣log∣log∣ϵdig​∣∣∣∼nQ​ (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⁡∣ϵlatt∣∣∼NQ|\log |\epsilon_{\rm latt}||\sim N_Q∣log∣ϵlatt​∣∣∼NQ​ (total number of qubits in the simulation). We find that fewer than nQ=10n_Q =10nQ​=10 qubits per spatial lattice site are sufficient to reduce digitization errors below noise levels expected in NISQ-era quantum devices for both localized and delocalized field-space wavefunctions. For localized wavefunctions, nQ=4n_Q = 4nQ​=4 qubits are likely to be sufficient for calculations requiring modest precision.
Quantum simulation of quantum field theory is a flagship application of quantum computers that promises to deliver capabilities beyond classical computing. The realization of quantum advantage will require methods that can accurately predict error scaling as a function of the resolution and parameters of the model and that can be implemented efficiently on quantum hardware. In this paper, we address the representation of lattice bosonic fields in a discretized field amplitude basis, develop methods to predict error scaling, and present efficient qubit implementation strategies. A low-energy subspace of the bosonic Hilbert space, defined by a boson occupation number cutoff, can be represented with exponentially good accuracy by a low-energy subspace of a finite-size Hilbert space. The finite representation construction and the associated errors are directly related to the accuracy of the Nyquist-Shannon sampling and the finite Fourier transforms of the boson number states in the field and the conjugate-field bases. We analyze the relation between the boson mass, the discretization parameters used for wave function sampling, and the finite representation size. Numerical simulations of small size ${\mathrm{\ensuremath{\Phi}}}^{4}$ problems demonstrate that the boson mass optimizing the sampling of the ground state wave function is a good approximation to the optimal boson mass yielding the minimum low-energy subspace size. However, we find that accurate sampling of general wave functions does not necessarily result in accurate representation. We develop methods for validating and adjusting the discretization parameters to achieve more accurate simulations.
Initializing a single site of a lattice scalar field theory into an arbitrary state with support throughout the quantum register requires O(2nQ)O({2}^{{n}_{Q}})O(2nQ​) entangling gates on a quantum computer with nQ{n}_{Q}nQ​ qubits per site. It is conceivable that instead initializing to functions that are good approximations to states may have utility in reducing the number of required entangling gates. In the case of a single site of a noninteracting scalar field theory, initializing to a symmetric exponential wave function requires ${n}_{Q}\ensuremath{-}1$ entangling gates, the minimal number necessary to create an entangled state of all nQ{n}_{Q}nQ​ qubits. This is compared with the ${2}^{{n}_{Q}\ensuremath{-}1}+{n}_{Q}\ensuremath{-}3+{\ensuremath{\delta}}_{{n}_{Q},1}$ required for a symmetric Gaussian wave function. In this work, we explore the initialization of one-site (nQ=4{n}_{Q}=4nQ​=4), two-site (nQ=3{n}_{Q}=3nQ​=3), and three-site (nQ=3{n}_{Q}=3nQ​=3) noninteracting scalar field theories with symmetric exponential wave functions using IBM's quantum simulators and quantum devices (Poughkeepsie and Tokyo). With the digitizations attainable with nQ=3,4{n}_{Q}=3,4nQ​=3,4, these tensor-product wave functions are found to have large overlap with a Gaussian wave function and provide a suitable low-noise initialization for improvement and Somma inflation. In performing these simulations, we have employed a workflow that interleaves calibrations to mitigate systematic errors in production. The calibrations allow tolerance cuts on gate performance including the fidelity of the symmetrizing Hadamard gate, both in vacuum (${|\mathbf{0}\ensuremath{\rangle}}^{\ensuremath{\bigotimes}{n}_{Q}}$) and in medium (${n}_{Q}\ensuremath{-}1$ qubits initialized to an exponential function). The results obtained in this work are relevant to systems beyond scalar field theories, such as the deuteron radial wave function, two- and three-dimensional Cartesian-space wave functions, and nonrelativistic multinucleon systems built on a localized eigenbasis.
In the near-future noisy intermediate-scale quantum era of quantum computing technology, applications of quantum computing will be limited to calculations of very modest scales in terms of the number of qubits used. The need to represent numeric quantities using limited resources leads to digitization errors which must be taken into account. As a first step towards quantum simulations of realistic high-energy physics problems, we use ensembles of SU(2) lattice gauge fields generated with standard classical computing to explore the effects of digitizing elements of the gauge group to a finite set. We consider several methods for digitizing the group, finding the best performance from an action-preserving projection onto a mesh. Working in (3+1)(3+1)(3+1) dimensions, we find that using $\ensuremath{\sim}7$ (qu)bits to represent each SU(2)\mathrm{SU}(2)SU(2) gauge link induces a digitization error of the order of 1010%10 in short-distance observables and 22%2 in long-distance observables. Promisingly, our results indicate that each SU(2)\mathrm{SU}(2)SU(2) gauge link can be represented by O(10)O(10)O(10) (qu)bits, from which we estimate that a 163SU(2){16}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(2)163SU(2) lattice could be simulated with no more than O(105)O({10}^{5})O(105) (qu)bits. Our results on digitization are also of interest as a form of lossy compression that could be used in high-performance classical computing to alleviate communications bottlenecks.
Quantum simulations of scalar quantum field theories (QFT) provide important benchmarks for demonstrating quantum advantage. We revisit digitization in the occupation basis, which is typically hindered by unfavorable circuit depth scaling. We present an approach that achieves exponential reductions in circuit depth and significantly mitigates Trotter errors by diagonalizing field operators prior to their decomposition into Pauli strings. Focusing on a scalar QFT in 2+1 dimensions, we show that this method substantially reduces circuit depth and CNOT gate counts for time evolution. Using the Lorentzian energy-energy correlator as a benchmark observable, we find parameter regimes in which occupation-basis digitization converges more rapidly with respect to local truncation than the amplitude-basis approach of Jordan, Lee, and Preskill. These results provide both algorithmic advances and phenomenological benchmarks for studies of light-ray observables on near-term quantum devices.
Renormalization group ideas and effective operators are introduced to efficiently prepare ground states of massive lattice field theories on digital quantum devices. This is accomplished with a systematic approximation through localized unitaries that removes an exponentially costly barrier in the spatial volume of the quantum simulation. With these methods, classically computed ground states in a spatial volume LLL, containing a few Compton wavelengths, can be used to determine operators for preparing the ground state toward the thermodynamic limit with a precision improving as ${e}^{\ensuremath{-}mL}$ on beyond-classical quantum registers. Due to the exponential spatial decay of correlations in massive theories and the double exponential suppression of digitization artifacts in the number of qubits representing the scalar field, the derived fixed-point quantum circuits are expected to be relevant for simulations of quantum field theories throughout the evolution from small-scale near-term quantum devices to large-scale fault-tolerant quantum computers.