Quantum Imaginary Time Evolution Algorithm for Quantum Field Theories with Continuous Variables

Quantum fields are fundamental constituents of the physical world describing quantum many-body systems of matter at all energy scales, as well as electromagnetic and gravitational radiation. Quantum-field engineering has enabled unprecedented measurement sensitivities, epitomized by the use of squeezed light to lower the noise floor of the Laser Interferometer Gravitational-wave Observatory (LIGO) below the shot noise limit Aasi2013 .

The encoding of quantum information in continuous-variable (CV) quantum fields, a.k.a.  qumodes (in lieu of discrete-variable (DV) qubits), has enabled multipartite entanglement over millions of qumodes. This scale, unparalleled in any qubit architecture, defines new horizons and paradigms for quantum computing, quantum communication, and quantum sensing. Nanophotonic integrated devices based on qumodes have the potential to define future quantum technology by surpassing the performance of qubit-based Noisy Intermediate-Scale Quantum (NISQ) Preskill2018 computing devices.

A natural implementation of qumodes uses quantum light, which also lends itself to sensing Xia2021 ; Casacio2021 ; Pooser2020 ; Lawrie2019 and communication Xu2020 ; Pirandola2020 . The coming of age of low-loss, high-nonlinearity integrated optics paves the way for implementing large-scale, fault-tolerant quantum computing and communication devices on chip, at room temperature, and within a few years Sciarrino2020 .

Over the last 20 years, since the introduction of the first quantum algorithm by Deutsch and Jozsa Deutsch-Jozsa1992 , even though a tremendous amount of DV quantum algorithms has been proposed capable of solving various problems more efficiently than their classical counterparts, there has been far less activity in the development of CV quantum computing. Examples of quantum algorithms generalized to CV substrates include the Deutsch-Jozsa Adcock2009 and Grover’s search Pati2000 algorithms, and more recently the quantum approximate optimization algorithm (QAOA) Verdon2019 and a CV quantum algorithm for solving linear partial differential equations Arrazola2019 .

There has also been an effort to extend quantum machine learning algorithms to a CV substrate Lau2017 followed by the singular value decomposition of nonsparse, low rank matrices in Das2018 , and topological data analysis Siopsis2019 . CV quantum neural networks were introduced in Killoran2019 and used in applications such as fraud detection with a CV classifier, a hybrid classical-quantum auto encoder.

Considering the recent experimental breakthroughs in CV photonic quantum computing Zhong2020 and the development of a programmable photonic quantum computer chip by the Xanadu team Arrazola2021 , it is important to explore CV quantum algorithms applied to realistic experimental setups.

Here, we propose a CV quantum imaginary-time evolution (QITE) algorithm which can be used to calculate the energy levels and corresponding energy eigenstates of an interacting scalar quantum field theory (QFT) on a lattice. A QFT for a massive self-interacting scalar field $\phi$ with a $\phi^{4}$ interaction was studied in Jordan2012 where it was shown that quantum algorithms for scattering amplitudes provided an exponential speed up over known classical algorithms. QFTs were further studied in YeterAydeniz2019QFT ; Barata2021 ; Davoudi2021 with DV quantum algorithms and in Marshall2015 ; YeterAydeniz2018QED on a CV substrate for a $\phi^{4}$ scalar QFT as well as scalar quantum electrodynamics (QED). There is a growing interest in simulating lattice gauge theories using quantum computers as well Banuls2020 .

There have been a few alternative approaches. In Ref. Liu2020 , quantum computation of two-dimensional quantum chromodynamics (QCD) was discussed without introducing a lattice. A different perspective was offered in Milsted2021 where it was proposed that the noise of NISQ hardware can be avoided by simulating the dynamics of QFTs using classical simulators and tools such as Matrix Product States (MPS).

CV quantum algorithms for QFTs have important advantages over their DV counterparts. Their implementation generally relies on optical elements which operate at room temperature. Moreover, while DV algorithms need a whole register of qubits, only a single qumode is needed at each point on the lattice to simulate the field at that point. However, CV quantum algorithms that have been proposed so far involve non-Gaussian quantum gates which are hard to implement with existing technology bibCV1 ; bibCV2 ; bibCV3 ; bibCV4 . Here, we avoid the introduction of non-Gaussian gates by employing a CV adaptation of the QITE algorithm making use of detectors projecting onto eigenstates of the photon number operator which is readily available technology.

Imaginary-time evolution has been extensively used in studying quantum many-body systems as a useful tool for various tasks, such as the calculation of the ground state energy and the creation of finite-temperature states. Evolution in imaginary time $\tau$ is implemented with the non-unitary operator $\mathcal{U}(\tau)=e^{-\tau H}$ where $H$ is the Hamiltonian of the system of interest. Starting with an initial state that has non-zero overlap with the ground state of the system, the evolved state convergences to the ground state in the limit $\tau\to\infty$. Excited states can also be reached with an appropriate choice of initial state (one that is orthogonal to the ground state). The calculation of the energy spectrum of a many-body quantum system is a daunting yet important task as it provides important information about the system. As the number of particles increases, the calculation becomes exponentially harder. Then it becomes imperative to employ the quantum version of the imaginary-time evolution algorithm as it outperforms its classical counterpart.

Simulating the imaginary-time evolution on a quantum computer is not straightforward because $\mathcal{U}(\tau)$ is a non-unitary operator. Various approaches to the implementation of $\mathcal{U}(\tau)$ have been proposed. A variational version of QITE was proposed in McArdle2019 in which the Wick-rotated Schrödinger equation was solved for parameterized states and a classical optimization loop was used to estimate the parameters for the ground state of molecular Hydrogen and the LiH molecule. Although this method offers shallow quantum circuits, the classical optimization of the parameters becomes prohibitive as the number of particles in the system grows. Motta et al. Motta2019 proposed a QITE algorithm that did not require classical optimization or an ancilla qubit. They expressed the Hamiltonian in terms of local terms and used Trotterization to implement $\mathcal{U}(\tau)$. Then the non-unitary evolution operator for a small imaginary-time interval was approximated by a unitary operator which was expressed in terms of Pauli spin operators with coefficients calculated from measurements on quantum hardware. The problem with this method is that the number of measurements grows exponentially with the system size and the quantum circuit becomes longer with each QITE step making it hard to implement on NISQ devices. Various attempts to bypass this problem have been made YeterAydeniz2019 ; YeterAydeniz2020 ; Sun2020 ; Nishi2021 ; Shirakawa2021 .

We implement the QITE algorithm by approximating the non-unitary evolution operator with a Gaussian operator, $\mathcal{U}(\tau)\approx e^{-\tau\mathcal{A}}$, where $\mathcal{A}$ is a Hermitian operator and a function of one of the two quadratures, $q$. After Trotterization, the Ansatz $\mathcal{A}$ is determined at each step in a manner similar to the DV approach in Motta2019 . The non-unitary Gaussian operator $e^{-\tau\mathcal{A}}$ is realized with the aid of ancilla qumodes. Only a single ancilla is needed at each step regardless of the size of the system. As the number of steps increases, the length of the quantum circuit does not increase indefinitely, unlike in the DV case, because the contributions to the Gaussian operator $\mathcal{A}$ at each QITE step commute with each other. Thus the quantum circuit only involves a finite number of parameters which are determined by quantum measurements at each QITE step. Thus, we avoid the use of non-Gaussian elements and rely on quantum measurements that involve detectors projecting onto photon-number eigenstates. We explain how our quantum algorithm can be realized with existing technology.

Our discussion is organized as follows. In Section II, we review the discretization of the massive $\phi^{4}$ self-interacting scalar QFT. In Section III we discuss the details of our CV quantum algorithm. In Section IV, we discuss our results using Xanadu’s Strawberry Fields CV photonic quantum simulator Killoran2019SF ; Bromley2020 and show that they are in agreement with exact results. We also outline an experimental realization of the quantum circuit with existing technology. We present our conclusions and outlook in Section V.

For definiteness, we concentrate on the simplest QFT describing a massive self-interacting scalar field in one spatial dimension with a quartic interaction. Our results can be generalized to more complicated QFTs, including gauge theories that describe elementary particle interactions.

To study the QFT on a quantum computer, we discretize the system in space with coordinate $x=0,1,\dots,L-1$ where $L$ is the length of the spatial dimension in units in which the lattice spacing is $a=1$. The Hamiltonian for a massive scalar field $\phi(x)$ with a quartic interaction term is

where $m_{0}$ is the bare mass of the scalar field, $\lambda$ is the interaction strength, and $\pi(x)$ is the conjugate momentum obeying the commutation relations

The scalar field and its conjugate momentum can be expressed in terms of creation and annihilation operators obeying commutation relations $\left[a(k),a^{\dagger}(k^{\prime})\right]=\delta_{kk^{\prime}}$ as

where

The mass parameter $m$ is arbitrary as long as $m^{2}>0$. One may choose $m=m_{0}$, but this is often not possible in physically interesting cases in which $m_{0}^{2}<0$. If $m$ is chosen as the physical mass parameter, $m=m_{\text{phys}}$ (notice that in an interacting system $m_{\text{phys}}\neq m_{0}$, due to quantum effects), then in the continuum limit, $\omega(k)$ is the energy of a relativistic particle of momentum $\frac{2\pi k}{L}$. No physical quantities depend on the choice of $m$, although spurious dependencies enter if one simulates the system with qubits due to the truncation of the Hilbert space. This is not an issue with CV quantum algorithms.

Having chosen $m$, we define the mass counter term $\delta m$ by $m^{2}=m_{0}^{2}+\delta m$. The Hamiltonian splits into a non-interacting ($H_{0}$) and interacting ($H_{I}$) piece,

where

The non-interacting Hamiltonian is diagonal in the momentum representation,

whereas the interaction part is diagonal in the position representation.

For the CV quantum calculation, it is convenient to work with the quadratures

in terms of which the non-interacting Hamiltonian reads

Its ground state can be written as

where $\bm{q}=(q(0),\dots,q(L-1))$. It has zero energy, by design.

Alternatively, we can work with the quadratures in the position representation, $\phi(x),\pi(x)$. The two sets of quadratures are related to each other by a Bogoliubov transformation that can be implemented with beam splitters and squeezers,

In this section, we introduce our CV quantum algorithm for the calculation of energy levels and corresponding eigenstates of the scalar QFT.

In this section, we present our simulator results for the calculation of ground and excited state energies using our CV QITE algorithm. The quantum circuits depicted in Figures 1 and 2 create approximations to the ground and first-excited states, respectively, as outlined above. Even though these figures are for $L=1$-point case the generalization to $L$-point case is straightforward. These circuits can be easily simulated with Xanadu’s Strawberry Fields photonic quantum computer simulator where the required states, Gaussian gates, and measurement tools are readily available. They can also be realized experimentally with existing technology.

The energy expectation value obtained at each QITE step is plotted in Fig. 3 as a function of imaginary time for the single-point lattice, $L=1$, a toy model. We compare the values obtained from the simulator to exact analytic calculations for strength of interaction $\lambda=4.8$, cutoff for the Hilbert space dimension in the simulator $n_{\text{cutoff}}=20$, and $CX$ gate parameter $\eta=0.1$. Starting the CV QITE algorithm with initial state $|\Psi[0]\rangle=|\Omega_{0}\rangle$, the energy expectation values are seen to converge to the ground state energy. Due to the cutoff in the Hilbert space dimension needed due to the limitations of the simulator, there is a 1.75% error in the ground state energy.

As discussed above, by choosing one of the odd wavefunctions $|\Omega(k)\rangle$ (Eq. (33)) as initial state, the QITE algorithm approximates excited states of the system. The quantum algorithm in this case relies on the experimental setup depicted in Fig. 2 which can also be realized experimentally with existing technology.

In Fig. 4, we demonstrate the convergence of the energy expectation values to the ground, first excited, and third excited state energies, starting with initial states $|\Omega_{0}\rangle$, $|\Omega(0)\rangle$, and $|\Omega(1)\rangle$, respectively, for the case of two lattice points ($L=2$). We used Xanadu’s Strawberry Fields photonic simulator to obtain the simulator results. The parameters used were $m^{2}=0.1$, $\lambda=1$, and $\eta=0.1$. We ran the CV quantum circuit in Fig. 1 at each QITE step for initial state $|\Omega_{0}\rangle$, and the one in Fig. 2 for initial states $|\Omega(k)\rangle$ ($k=0,1$). The calculation of the various terms contributing to the parameters $\gamma_{s}(k)$ (Eq. (20)) require additional measurements which were done with the aid of quantum circuits depicted in Fig. 2. In these calculations we only used the contributions coming from $\gamma_{s}(0)$ since it is the most contributing to the convergence of the energy expectation value. For this reason, the generalization of the quantum circuit to $L=2$-point case is just repetition of circuit in Fig. 1 and Fig. 2 except an initial squeezer in the second qumode. We calculated the derivatives needed (Eq. (31)) using the differential quadrature method Bert1996 . We used equal spacing of 0.1 between parameters in the derivative calculation in the $CX$ gate. It should be noted that the calculation of high-order derivatives requires a certain precision in the parameters of the $CX$ gate (Eq. (23)), the squeezing parameter $r$ and the beam splitter angle $\theta$, as well the measured probability $P_{0}(k)$ (Eq. (30)). In Fig. 6, we show the relative uncertainty $\frac{\Delta d_{3}}{d_{3}}$ in the value of the third derivative $d_{3}=\frac{d^{3}P_{0}}{d(\eta^{2})^{3}}$ in terms of the uncertainty on the squeezing parameter $r$, for sample spacing $\eta^{2}_{i+1}-\eta^{2}_{i}$ values. Although we observe a sensitivity in the third derivative, we still achieve low relative uncertainty with reasonable sample spacing which can be further reduced by selecting appropriate, unevenly spaced samples.

Fig. 5 depicts the mass gap for the two-point lattice model ($L=2$) with parameters $m_{0}^{2}=-0.1$, $\delta m=0.2$, $\lambda=1$. It converges rapidly to the expected value.

In this work, we developed a CV QITE algorithm to calculate the ground- and excited-state energies of a $\phi^{4}$ scalar QFT on a one-dimensional spatial lattice. Unlike with DV quantum computing, where a register of qubits is needed at each lattice point and the Hilbert space is truncated, only a single qumode at each lattice point was needed. The algorithm required ancilla qumodes for its implementation. The ancilla qumodes did not add a significant overhead as their number was comparable to the number of qumodes needed for the system. Our CV quantum algorithm avoided the use of non-Gaussian gates which are technologically challenging to implement. It only relied on Gaussian operators and measurements projecting onto photon number eigenstates. Therefore, it can be implemented with existing technology. Figures 1 and 2 show the CV quantum circuits involved.

We implemented our algorithm in simple cases of lattices with $L=1$ (toy model) and $L=2$ points using Xanadu’s Strawberry Fields photonic quantum simulator. We observed convergence of energy expectation values to the ground- and excited-state energies depending on the choice of initial state, $|\Omega_{0}\rangle$ (Eq. (11)) and $|\Omega(k)\rangle$ (Eq. (33)), respectively. Higher energy levels can also be reached by using the states derived in CV QITE as input to a CV version of the QLanczos algorithm.

Our results provide the basis for the development of CV quantum algorithms that rely on Gaussian operators and photonic measurements which can be implemented with existing technology. Our method can be extended to QFTs such as gauge theories that describe elementary particle interactions and the states derived with our CV QITE algorithm can be used for the calculation of various physical quantities of interest, such as scattering amplitudes. Work in this direction is in progress.