arXiv:1912.03577 · Physical Review A

Systematically localizable operators for quantum simulations of quantum field theories

Natalie Klco, Martin J. Savage A

Published Jul 22, 2020• 36 citations

Abstract

Correlations and measures of entanglement in ground-state wave functions of relativistic quantum field theories are spatially localized over length scales set by the mass of the field. We utilize this localization to design digital quantum circuits for preparing the ground states of lattice scalar quantum field theories. Controlled rotations that are exponentially localized in their position-space extent are found to provide exponentially convergent wave function fidelity. These angles scale with the correlation between sites and the classical two-point correlation function, as opposed to the more localized mutual information or the hyperlocalized $(1\ifmmode\times\else\texttimes\fi{}1)$-site negativity. We anticipate that further investigations will uncover quantum circuit designs with controlled rotations dictated by the measures of entanglement. This work is expected to impact quantum simulations of systems of importance to nuclear physics, high-energy physics, and basic energy sciences research.

DOI: 10.1103/physreva.102.012619

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Entanglement and Correlations in Cold Atomic Systems

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Euclidean-Monte-Carlo-informed ground-state preparation for quantum simulation of scalar field theory

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arXiv:1912.03577 · Physical Review A

Systematically localizable operators for quantum simulations of quantum field theories

Natalie Klco, Martin J. Savage A

Published Jul 22, 2020• 36 citations

arXiv HTML PDF

Abstract

(1\ifmmode\times\else\texttimes\fi{}1)

-site negativity. We anticipate that further investigations will uncover quantum circuit designs with controlled rotations dictated by the measures of entanglement. This work is expected to impact quantum simulations of systems of importance to nuclear physics, high-energy physics, and basic energy sciences research.

DOI: 10.1103/physreva.102.012619

Related papers

Fixed-point quantum circuits for quantum field theories

Natalie Klco, Martin J. Savage A

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 $L$ , 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.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 $L$, 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.

arXiv:2002.02018Physical Review ANov 30, 202023 citations

Exponentially improved quantum simulation of scalar QFT

Qing-Hong Cao, Ying-Ying Li, Xiaohui Liu +2 more

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.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.

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Minimally entangled state preparation of localized wave functions on quantum computers

Natalie Klco, Martin J. Savage A

Initializing a single site of a lattice scalar field theory into an arbitrary state with support throughout the quantum register requires $O({2}^{{n}_{Q}})$ entangling gates on a quantum computer with ${n}_{Q}$ 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 ${n}_{Q}$ 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 ( ${n}_{Q}=4$ ), two-site ( ${n}_{Q}=3$ ), and three-site ( ${n}_{Q}=3$ ) 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 ${n}_{Q}=3,4$ , 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.Initializing a single site of a lattice scalar field theory into an arbitrary state with support throughout the quantum register requires $O({2}^{{n}_{Q}})$ entangling gates on a quantum computer with ${n}_{Q}$ 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 ${n}_{Q}$ 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 (${n}_{Q}=4$), two-site (${n}_{Q}=3$), and three-site (${n}_{Q}=3$) 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 ${n}_{Q}=3,4$, 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.

arXiv:1904.10440Physical Review AJul 17, 202067 citations

Digitization of scalar fields for quantum computing

Natalie Klco, Martin J. Savage A

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.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.

arXiv:1808.10378Physical Review AMay 23, 2019165 citations

Entanglement and Correlations in Cold Atomic Systems

M. Popp

This thesis is devoted to the theoretical characterization and generation of strongly correlated many-body states from the perspective of quantum information theory.We first study the entanglement properties of general many-body states and define a new measure for multipartite entanglement, the "Localizable Entanglement (LE)".It quantifies the maximal entanglement that can be localized, on average, between two parties by performing local measurements on the other parties.This measure provides knowledge of the "entanglement length", which is of particular importance in the context of quantum repeaters for long-distance quantum communication.We prove upper and lower bounds on LE in terms of two-body correlation functions and develop a method for the numerical computation of LE in one-dimensional spin systems.We analyze the localizable entanglement in various spin systems and observe characteristic features for a quantum phase transition such as a diverging entanglement length.In the second part of the thesis we focus on a specific many-body system: ultra-cold atoms in an optical lattice.This system is a promising candidate for the first physical realization of a "quantum simulator", but the temperature in current experiments is still too high.We propose, analyze and compare several schemes to cool the atoms close to the ground state of the strong interaction regime.In particular, we devise an algorithmic cooling protocol that combines occupation number filtering with spin-dependent lattice shifts.In addition, we propose two different physical realizations of filtering and also design protocols that generate an ensemble of quantum registers for quantum computation.In the third part we propose how to create entanglement in small atomic clouds by deforming and rotating the harmonic trapping potential.The resulting states are entangled in their motional degrees of freedom and can be identified with strongly correlated fractional quantum Hall states.For the case of two, three and four atoms we show how to adiabatically transform the unentangled ground state into the desired entangled state.We further discuss characteristic features of these states and propose how to create and detect them experimentally using an optical lattice setup. A Proof of the lower bound on LEA.1 Theorem on connected correlation functions and local measurements . . . . . . . . .This thesis is devoted to the theoretical characterization and generation of strongly correlated many-body states from the perspective of quantum information theory.We first study the entanglement properties of general many-body states and define a new measure for multipartite entanglement, the "Localizable Entanglement (LE)".It quantifies the maximal entanglement that can be localized, on average, between two parties by performing local measurements on the other parties.This measure provides knowledge of the "entanglement length", which is of particular importance in the context of quantum repeaters for long-distance quantum communication.We prove upper and lower bounds on LE in terms of two-body correlation functions and develop a method for the numerical computation of LE in one-dimensional spin systems.We analyze the localizable entanglement in various spin systems and observe characteristic features for a quantum phase transition such as a diverging entanglement length.In the second part of the thesis we focus on a specific many-body system: ultra-cold atoms in an optical lattice.This system is a promising candidate for the first physical realization of a "quantum simulator", but the temperature in current experiments is still too high.We propose, analyze and compare several schemes to cool the atoms close to the ground state of the strong interaction regime.In particular, we devise an algorithmic cooling protocol that combines occupation number filtering with spin-dependent lattice shifts.In addition, we propose two different physical realizations of filtering and also design protocols that generate an ensemble of quantum registers for quantum computation.In the third part we propose how to create entanglement in small atomic clouds by deforming and rotating the harmonic trapping potential.The resulting states are entangled in their motional degrees of freedom and can be identified with strongly correlated fractional quantum Hall states.For the case of two, three and four atoms we show how to adiabatically transform the unentangled ground state into the desired entangled state.We further discuss characteristic features of these states and propose how to create and detect them experimentally using an optical lattice setup. A Proof of the lower bound on LEA.1 Theorem on connected correlation functions and local measurements . . . . . . . . .

Jan 1, 20060 citations

Euclidean-Monte-Carlo-informed ground-state preparation for quantum simulation of scalar field theory

Navya Gupta, Christopher David White, Zohreh Davoudi

Quantum simulators offer great potential for investigating dynamical properties of quantum field theories. However, preparing accurate non-trivial initial states for these simulations is challenging. Classical Euclidean-time Monte-Carlo methods provide a wealth of information about states of interest to quantum simulations. Thus, it is desirable to facilitate state preparation on quantum simulators using this information. To this end, we present a fully classical pipeline for generating efficient quantum circuits for preparing the ground state of an interacting scalar field theory in 1+1 dimensions. The first element of this pipeline is a variational ansatz family based on the stellar hierarchy for bosonic quantum systems. The second element of this pipeline is the classical moment-optimization procedure that augments the standard variational energy minimization by penalizing deviations in selected sets of ground-state correlation functions (i.e., moments). The values of ground-state moments are sourced from classical Euclidean methods. The resulting states yield comparable ground-state energy estimates but exhibit distinct correlations and local non-Gaussianity. The third element of this pipeline is translating the moment-optimized ansatz into an efficient quantum circuit with an asymptotic cost that is polynomial in system size. This work opens the way to systematically applying classically obtained knowledge of states to prepare accurate initial states in quantum field theories of interest in nature.Quantum simulators offer great potential for investigating dynamical properties of quantum field theories. However, preparing accurate non-trivial initial states for these simulations is challenging. Classical Euclidean-time Monte-Carlo methods provide a wealth of information about states of interest to quantum simulations. Thus, it is desirable to facilitate state preparation on quantum simulators using this information. To this end, we present a fully classical pipeline for generating efficient quantum circuits for preparing the ground state of an interacting scalar field theory in 1+1 dimensions. The first element of this pipeline is a variational ansatz family based on the stellar hierarchy for bosonic quantum systems. The second element of this pipeline is the classical moment-optimization procedure that augments the standard variational energy minimization by penalizing deviations in selected sets of ground-state correlation functions (i.e., moments). The values of ground-state moments are sourced from classical Euclidean methods. The resulting states yield comparable ground-state energy estimates but exhibit distinct correlations and local non-Gaussianity. The third element of this pipeline is translating the moment-optimized ansatz into an efficient quantum circuit with an asymptotic cost that is polynomial in system size. This work opens the way to systematically applying classically obtained knowledge of states to prepare accurate initial states in quantum field theories of interest in nature.

arXiv:2506.02313Physical review. D/Physical review. D.Jan 5, 20262 citations

(1\ifmmode\times\else\texttimes\fi{}1)

L

, 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.

Initializing a single site of a lattice scalar field theory into an arbitrary state with support throughout the quantum register requires

O({2}^{{n}_{Q}})

entangling gates on a quantum computer with

{n}_{Q}

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

{n}_{Q}

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 (

{n}_{Q}=4

), two-site (

{n}_{Q}=3

), and three-site (

{n}_{Q}=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

{n}_{Q}=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.

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.