 Original article
 Open Access
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Constantdepth circuits for dynamic simulations of materials on quantum computers
Materials Theory volume 6, Article number: 13 (2022)
Abstract
Dynamic simulation of materials is a promising application for nearterm quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to shorttime dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a specific subset of onedimensional (1D) materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable timesteps, the constantdepth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many timesteps. For an Nspin system, the constantdepth circuit contains only \(\mathcal {O}(N^{2})\) CNOT gates. Such compact circuits enable us to successfully execute longtime dynamic simulation of ubiquitous models, such as the transverse field Ising and XY models, on current quantum hardware for systems of up to 5 qubits without the need for complex error mitigation techniques. Aside from enabling longtime dynamic simulations with minimal Trotter error for a specific subset of 1D Hamiltonians, our constantdepth circuits can advance materials simulations on quantum computers more broadly in a number of indirect ways.
Introduction
Quantum computers are intrinsically fit for efficiently simulating quantum systems (Feynman 1982; Lloyd 1996; Abrams and Lloyd 1997; Zalka 1998), making the simulation of quantum materials a leading “killer application” for this novel technology. Nearterm quantum computers, also known as noisy intermediatescale quantum (NISQ) computers, suffer from short qubit decoherence times and high gate error rates, making it difficult to achieve highfidelity results with large quantum circuits (Preskill 2018). Thus, one of the major challenges with performing simulations on current quantum hardware is keeping the circuits small enough such that their results remain distinguishable from random noise.
This is particularly challenging for dynamic simulations, which require the execution of one circuit per timestep, where each circuit implements the timeevolution operator from the initial time to the given timestep (Bassman et al. 2021). Current algorithms for dynamic materials simulations produce quantum circuits whose depths grow with increasing timestep count (Wiebe et al. 2011; Childs et al. 2018). Thus, an essential part of the workflow is quantum circuit optimization, which aims to minimize the depth of the circuits. Already, a great deal of research has focused on general circuit optimization (i.e. minimization) (Möttönen et al. 2004; De Vos and De Baerdemacker 2016; Iten et al. 2016; Martinez et al. 2016; Khatri et al. 2019; Murali et al. 2019; Younis et al. 2021; Cincio et al. 2020), which is an NPhard problem (Botea et al. 2018; Herr et al. 2017). More recently, domainspecific circuit optimizers, which focus on optimizing certain types of circuits for specific applications, have been suggested (Bassman et al. 2020) as a method to reduce to complexity of this optimization problem.
According to the “nofastforwarding theorem”, simulating the dynamics of a system under a generic Hamiltonian H for a time t requires Ω(t) gates (Berry et al. 2007; Childs and Kothari 2010), implying that circuit depths grow at least linearly with the number of timesteps. It has been shown, however, that quadratic Hamiltonians can be fastforwarded, meaning the evolution of the systems under such Hamiltonians can be simulated with circuits whose depths do not grow significantly with the simulation time (Atia and Aharonov 2017; Gu et al. 2021). A recent work took advantage of this to variationally compile approximate circuits with a hybrid classicalquantum algorithm for fastforwarded simulations (Cîrstoiu et al. 2020). The circuits, however, are approximate, with error that grows with increasing fastforwarding time.
Here, we present an algorithm for generating quantum circuits that are constant in depth with increasing timestep count for simulations of materials governed by a specific set of models derived from the onedimensional (1D) Heisenberg Hamiltonian, which we denote as \(\mathcal {H_{CD}}\) and define in Theoretical background section. This set \(\mathcal {H_{CD}}\), whose most prominent models include the transverse field Ising model (TFIM) and the (transverse field) XY model, is characterized by Hamiltonians that can be mapped to free fermionic models. While such models are known to be classically simulatable with polynomial resources (Valiant 2002; Terhal and DiVincenzo 2002), our constantdepth circuits can nonetheless help advance current research into materials simulations on nearterm quantum computers. This is illustrated by the ubiquity of these models in current research on nearterm quantum computers (Zhukov et al. 2018; Lamm and Lawrence 2018; Gustafson et al. 2019; Zhu et al. 2020; Bassman et al. 2020; YeterAydeniz et al. 2021; Bassman et al. 2021; Sun et al. 2021; Neill et al. 2021). Indeed, systems like the TFIM are quintessential in the study of quantum phase transitions (Suzuki et al. 2012; GómezRuiz et al. 2016; Yang et al. 2019), ergodicity (Cheraghi and Mahdavifar 2020), critical behavior (Granato 1992), as well as myriad condensed matter systems, such as ferroelectrics (Blinc et al. 1979) and magnetic spin glasses (Wu et al. 1991). When these models are made timedependent, nonequilibrium effects such as dynamic phase transitions and quantum hysteresis can be studied (Tomé and de Oliveira 1990; Acharyya and Chakrabarti 1995; Acharyya 1998; Sides et al. 1998).
The constantdepth circuits for simulating the dynamics of models in \(\mathcal {H_{CD}}\) are comprised of twoqubit gates, known as matchgates (Valiant 2002). While generic twoqubit gates decompose into nativegate circuits with at most three CNOT gates (Vidal and Dawson 2004), the matchgates in our constantdepth circuits only require two CNOT gates in their decomposition. This special property allows us to introduce a set of conjectured matchgate identities, which enable the downfolding of our circuits for dynamic simulations into constantdepth for any number of timesteps.
The circuits have a fixed structure, with only the singlequbit rotation angles changing with the addition of more timesteps. The structure has a regular pattern which can be easily extrapolated to build circuits for any system size; for an Nspin system, the circuit structure contains only N(N−1) CNOT gates. Furthermore, the circuits are exact up to Trotter error, which we argue, can be practically eliminated. This is because Trotter error scales with the size of the simulation timestep, and the constantdepth nature of the circuits allows for a simulation to be feasibly broken into arbitrarily many (i.e., arbitrarily small) timesteps. For a given system size, if the constantdepth circuit is small enough to achieve highfidelity results on a NISQ computer, the dynamics of that system can be successfully simulated out to arbitrarily long times and with arbitarily small Trotter error.
Theoretical background
The quantum circuits for dynamic simulations of quantum materials must implement the timeevolution operator between the initial time (which we set to 0) and some final time t, given by
where \(\mathcal {T}\) indicates a timeordered exponential and H(t) is the timedependent Hamiltonian of the material. In general, this operator is challenging to compute exactly due to the timedependence of the Hamiltonian and the exponentiation of the Hamiltonian. Typically, the Trotter decomposition (Trotter 1959) is used to approximately construct U(t). With this method, the timedependent Hamiltonian H(t) is first approximated as a piecewise constant function by discretizing time into small timesteps over which H(t) is constant (Poulin et al. 2011). Next, the Hamiltonian at each timestep is split into components that are each individually easy to diagonalize, which enables Trotter decomposition to be performed at each timestep. In this way, the timeevolution operator is approximated as:
where τ multiplies over the number of discretized timesteps Δt and l multiplies over the components into with H(t) was divided. We note that other techniques for approximating the unitary operator exist (Childs and Wiebe 2012; Chen et al. 2021), but are rarely used in practice at present as the circuits they produce are far too large for current hardware.
The error generated from the Trotter decomposition, known as Trotter error, can be a significant source of error, scaling with the size of the simulation timestep Δt. Dynamic simulations based on Trotter decomposition must therefore strike a balance when selecting the size of Δt. This is because standard algorithms for such simulations produce circuits which grow in depth with increasing numbers of timesteps, which in turn limits the number of timesteps that are feasible to simulate to just a handful (Smith et al. 2019). While decreasing Δt will lower Trotter error, making Δt too small will not allow for a long enough total simulation time, since the number of timesteps is limited. Our constantdepth circuits, however, remove the limitation on the number of timesteps that can be feasibly simulated, since the circuits do not get deeper with higher timestep count. This allows for the timestep Δt to be made arbitrarily small, which in turn allows one to decrease the Trotter error to negligible values. Such practical elimination of Trotter error with constantdepth circuits can enable far more accurate simulation results for longtime dynamic simulations.
The constantdepth circuits we introduce here simulate the dynamical evolution of a quantum material whose Hamiltonian is a simplified version of the 1D Heisenberg model, as explained below. The Heisenberg Hamiltonian is defined as
where α sums over {x,y,z}, the coupling parameters J_{α} denote the exchange interaction between nearestneighbor spins along the αdirection, \(\sigma _{i}^{\alpha }\) is the αPauli matrix acting on qubit i, and h_{β}(t) is the timedependent amplitude of an external magnetic field along the βdirection, where β∈{x,y,z}. This Hamiltonian is thus defined by the set of its parameters {J_{x},J_{y},J_{z},h_{β}(t)}. We denote the set of all parameter sets as \(\mathcal {H}\). The full Heisenberg model is obtained when all parameters in the set are nonzero, however a number of ubiquitous models can be derived by setting various parameters to zero.
Table 1 shows all subsets \(\mathcal {H_{CD}}\) of \(\mathcal {H}\) for which we find that our constantdepth circuits work. The rows of the table denote either the direction of the external magnetic field h_{β} or a lack of field, while the columns label which of the coupling parameters are nonzero. The first three columns denote parameter sets where one coupling term is nonzero, the next three columns denote sets where two coupling terms are nonzero, while the final column denotes the sets where all three coupling parameters are nonzero. An × appears in table entries for parameter sets that define Hamiltonians in \(\mathcal {H_{CD}}\), which can be simulated with our constantdepth circuits. Note that J_{x}·J_{y}·J_{z}=0 is a necessary but not sufficient condition for constantdepth. We remark that all Hamiltonians in \(\mathcal {H_{CD}}\) can be mapped to free fermionic models.
As all \(\mathcal {H_{CD}}\) Hamiltonians of Table 1 are quadratic, it is possible to fastforward simulations under their timeevolution (Atia and Aharonov 2017; Gu et al. 2021). In Demonstration of constantdepth circuits section, we demonstrate simulations with our constantdepth circuits for two important models in \(\mathcal {H_{CD}}\): (i) the XY model, where J_{z}=0 and h_{β}=0, and (ii) the TFIM, where J_{y}=J_{z}=0. Dynamics of these models have recently been simulated on quantum computers, but lack of constantdepth circuits limited the number of timesteps that could be successfully simulated (Smith et al. 2019).
Construction of constantdepth circuits
To arrive at the circuit structure for the constantdepth circuits, we begin by laying down the gates that implement evolution of the system by one timestep, U(Δt). Due to the quadratic nature of \(\mathcal {H_{CD}}\) Hamiltonians, which only contain coupling interactions between nearest neighbor spins, the circuit for evolution of one timestep can be constructed by a set of twoqubit gates which act on each of the pairs of nearest neighbor qubits. For example, for six qubits, this circuit is given by
where each gate labeled G(Θ_{i}) is a twoqubit gate defined by some set of parameters Θ_{i}. For ease of notation, the parameter set Θ_{i} is dropped in subsequent labeling of these gates, which will simply be labeled with a G. However, it must be emphasized that each twoqubit gate has its own unique parameter set. Each additional timestep requires one additional repetition of the circuit for one timestep. In this manner, it is possible to construct circuits for dynamic simulations that grow with increasing numbers of timesteps. We refer to these circuits as “growing depth circuits” for dynamic simulations. The growing depth circuit for n timesteps for six qubits is thus given by
where there are 2n columns of G gates for n timesteps. We now show that it is possible to reduce the growing depth circuits for higher numbers of timesteps down to a constantdepth circuit which is comprised of N columns of G gates for an Nspin system, where each column alternates placing the top of the first G gate on the first or second qubit. The ability to “downfold” longer circuits into constantdepth circuits is derived from special properties of these G gates.
In fact, the G gates belong to a special group of twoqubit gates known as matchgates (Valiant 2002).
Definition 1
Let the matrices A and B be in SU(2)
with det(A)= det(B). Then the twoqubit matchgate G(A,B) is defined as follows
Matchgates have the important property that the product of two matchgates is again a matchgate and this will be a key feature to arrive at constantdepth circuits.
Lemma 1
Let G(A_{1},B_{1}) and G(A_{2},B_{2}) be matchgates, then the matrix
is again a matchgate with A_{3}=A_{1}A_{2} and B_{3}=B_{1}B_{2}.
Proof
The proof directly follows from carrying out the matrixmatrix multiplication. □
A graphical representation of Lemma 1 is given by
The decomposition of a general matchgate into a nativegate circuit, which can be executed on NISQ devices, requires three CNOT gates (Vidal and Dawson 2004). However, all the matchgates for \(\mathcal {H_{CD}}\) have a special structure which allows them to be decomposed into nativegate circuits with only two CNOT gates. In \(\mathcal {H_{CD}}\) cases with an external magnetic field along the x or ydirections, the gates G in (4) do not have the matchgate structure but are spectrally equivalent with a matchgate and are in fact matchgates up to some π/2 rotations. Matchgates and their corresponding decomposition into nativegate circuits with twoCNOTs are given for all Hamiltonians in \(\mathcal {H_{CD}}\) in Appendix A.
The ability to contract the circuits to constant depth relies on an identity that we conjecture for these special \(\mathcal {H_{CD}}\) matchgates.
Conjecture 1
Let G_{1},G_{2},G_{3} be matchgates of a certain type in \(\mathcal {H_{CD}}\), then there exist three corresponding matchgates G_{4},G_{5},G_{6} of the same type so that
A graphical representation of Conjecture 1 is given by
Using numerical optimization to identify circuit parameters on either side of the equality, we empirically find this conjecture to be true for all trials where the circuits are comprised of \(\mathcal {H_{CD}}\) matchgates. It has, however, proven challenging to analytically compute the parameters of the circuit on the righthand side given the parameterized circuit on the lefthand side and vice versa. As a result, compilation of our constantdepth circuits requires numerical optimization to obtain circuit parameters. We emphasize that the equivalence (11) only holds for \(\mathcal {H_{CD}}\) matchgates, whereas the equivalence (9) holds for all matchgates.
Based on Eqs. 9 and 11 we can derive identities for higher numbers of qubits, where a set of N columns of matchgates across N qubits can be replaced by its mirror image, albeit with altered parameter sets for all the constituent matchgates. A demonstration of deriving the identity for 4qubits is shown in Appendix B. We refer to these identities as the matchgate mirroring identities. These conjectured identities are depicted in Fig. 1 for four and five qubits. Note that for even numbers of qubits the mirroring is about a vertical axis (Fig. 1a), while for odd numbers of qubits the mirroring is about a horizontal axis (Fig. 1b). We emphasize that the matchgate parameters are different on either side of the equality signs.
To understand how these mirroring identities allow for the construction of constant depth circuits, we notice that for an Nqubit growing depth circuit (e.g., Eq. 5) we can apply the matchgate mirroring identity to the last N columns of matchgates in the circuit. We note that applying this identity will change the parameters defining each matchgate within the mirroring group. Application of this identity will result in pairs of adjacent matchgates on the same qubit pairs that can be combined into one matchgate, thus reducing the number of columns of matchgates in the circuit by one. This can be repeated until only N columns of matchgates in the circuit remain. This process is demonstrated for six qubits in Fig. 2. Figure 2a shows the growing depth circuit for six qubits simulating n timesteps with the last six columns of matchgates in the circuit highlighted with an outline. Figure 2b shows one application of the matchgate mirroring identity for six qubits to these last six columns of matchgates. Note how after applying the identity, two pairs of matchgates emerge adjacent to one another on the same pair of qubits, highlighted with an outline. These pairs can each be merged into one matchgate with new parameters, thus reducing the number of columns of matchgates in the circuit by one. This process is repeated until only six columns of matchgates remain, as shown in Fig. 2c.
The downfolding approach presented in Fig. 2 shows how to methodically obtain constantdepth circuits for each timestep in the dynamic simulation. In practice, however, we directly use numerical optimization to find the parameters for the constantdepth circuit of Fig. 2c. We begin by computing the operator in Eq. 2 (either by exact diagonalization or Trotter decomposition), which defines our target matrix (i.e., the matrix our circuit aims to carry out). Given a system size, we then construct the constantdepth circuit structure, which has N columns of matchgates for an Nqubit system. Next, we compute the matrix equivalent of the circuit, which will be compared to our target matrix. Using numerical optimization, we then solve for the parameters of the circuit that minimize the distance between the circuit matrix and the target matrix.
The number of circuit parameters grows quadratically with system size. This makes scaling to larger system sizes challenging as the circuit optimization for each timestep will take longer to compute. This could be ameliorated by finding a way to map the coefficients of the Hamiltonian directly to the rotation angles in the constantdepth circuit, whether through analytical techniques or machine learning methods. This would enable one to skip computation of the timeevolution operator and numerical optimization altogether. It should be noted, however, that the inability to remove this classical optimization step may not completely inhibit this method because the constantdepth circuit generation is embarrassingly parallel. In other words, the circuits for each timestep may all be computed in parallel, as numerical optimization for one circuit does not depend on information from any other circuit. In this way, the numerical optimization of circuits for all timesteps for large system simulations could be executed simultaneously on a classical supercomputer, which are regularly used for similar computations.
The circuit volume of the constantdepth circuits grows quadratically with system size N, while the depth grows only linearly with N. We emphasize, however, that unlike previous circuit generation techniques, our circuits do not grow in size with increasing numbers of timesteps, but rather remain fixed for a given system size N. This remarkable feature is what enables simulation out to arbitrarily large numbers of timesteps and thus permits longtime dynamic simulations. Most other methods for circuit generation will produce circuits that grow linearly with increasing numbers of timesteps (Wiebe et al. 2011; Smith et al. 2019). This prohibits dynamic simulations beyond a certain number of timesteps due to the quantum computer encountering circuits that are too large, and thus accumulate too much error due to gate errors and qubit decoherence.
Demonstration of constantdepth circuits
To demonstrate the power of our constantdepth circuits, we simulate quantum quenches of 3, 4, and 5spin systems defined by the TFIM and XY model on the IBM quantum processor “ibmq_athens”. A quantum quench is simulated by initializing the system in the ground state of an initial Hamiltonian, H_{i}, and then evolving the system through time under a final Hamiltonian, H_{f}. Quenches can simulate a sudden change in a system’s environment and provide insights into the nonequilibrium dynamics of various quantum materials.
The TFIM is obtained by setting J_{y}=J_{z}=0 and β=z in the Hamiltonian in Eq. 3. To perform a quench with the TFIM, we assume the external magnetic field is initially turned off, and the qubits are initialized in the ground state of an initial Hamiltonian \(H_{i}(t < 0) = \sum _{i} J_{x}\ \sigma _{i}^{x}\sigma _{i+1}^{x}\), which is a ferromagnetic state oriented along the xaxis. At time t=0, a timedependent magnetic field is instantaneously turned on, and the system evolves under the final Hamiltonian \(H_{i}(t \ge 0) = \sum _{i} \{J_{x}\sigma _{i}^{x}\sigma _{i+1}^{x} + h_{z}(t)\sigma _{i}^{z}\}\), which represents the TFIM. We use parameters from Ref Bassman et al. (2020), setting J_{x}=11.83898 meV and h_{z}(t)=2J_{x} cos(ωt) with ω=0.0048 fs^{−1}, which simulates a simplified model of a Redoped monolayer of MoSe2 under laser excitation. A timestep of 3 fs is used in the simulations. Our observable of interest is the average magnetization of the system along the xaxis, given by \(m_{x}(t) = \frac {1}{N}\sum _{i} \langle \sigma _{i}^{x}(t)\rangle \).
The XY model is obtained by setting J_{z}=h_{β}=0 in the Hamiltonian in Eq. 3. To perform a quench with the XY model, we initially let J_{z}→∞ and approximate the initial Hamiltonian \(H_{i}(t<0) = C \sum _{i}\sigma _{i}^{z}\sigma _{i+1}^{z}\), where C is an arbitrarily large constant. The ground state of this Hamiltonian is the Néel state, defined as ψ_{0}〉=↑↓↑⋯↓〉. At time t=0, we instantaneously set J_{z}=0, and let J_{x}=J_{y}=−1.0 eV, giving a final Hamiltonian of \(H_{f}(t \geq 0) = \sum _{i}\{\sigma _{i}^{x}\sigma _{i+1}^{x} + \sigma _{i}^{y}\sigma _{i+1}^{y}\}\), which represents the XY model. A timestep of 0.025 fs is used in the simulations. Our observable of interest is the staggered magnetization of the system, which is related to the antiferromagnetic order parameter and given by \(m_{s}(t) = \frac {1}{N}\sum _{i} (1)^{i} \langle \sigma _{i}^{z}(t)\rangle \).
To generate the constantdepth circuits for our simulations, we rely on circuit optimization software provided by the circuit synthesis toolkit BQSKit (Berkeley Quantum Synthesis Toolkit 2021). This suite of software provides several packages which can be used to generate the constantdepth circuits. The user must provide the matrix representation for the timeevolution operator to be implemented along with the parameterdependent constantdepth circuit structure. The circuit synthesis software then proceeds to use numerical optimization to find the optimal parameters for the circuit. Tutorials including the full code for generating our constantdepth circuits using the BQSKit toolkit are included in the Supplemental Material (Bassman et al. 2021).
Figure 3 shows the simulation results for quenches of the TFIM (top row) and XY model (bottom row) for various system sizes performed on a real quantum processor. The magnetization for each timestep was average over 8192 shots. In the first three columns, the results from our constantdepth circuits (red) and growing depth circuits (green) are compared to the expected results computed with a noisefree quantum computer simulator (blue). We note that timestep size was chosen sufficiently small such that the noisefree simulator results are in complete agreement with numerically exact results computed with exact diagonalization, and thus serve as our “ground truth”. The growing depth circuits were generated from standard Trotter decomposition and optimized with the IBM native compiler. While some new techniques have been developed for making shorter circuits based on Trotterization (Childs et al. 2018; Campbell 2019; Tran et al. 2020; Kivlichan et al. 2020; Childs et al. 2021), they nonetheless still grow in size with increasing timestep count, and thus will still generate results indistinguishable from random noise beyond a certain timestep. For this reason, the growing depth circuits produce qualitatively consistent results for the first few timesteps, but thereafter the circuits are too large, accumulating too much error, to produce highfidelity results. A recent benchmark study of dynamic simulations of similar systems on quantum computers found analogous behavior, with highfidelity results limited to only a handful of timesteps (Smith et al. 2019). In contrast, the results from our constantdepth circuits remain accurate for all timestep counts, and in principle, will remain so out to arbitrarily many timesteps. These results thus show the power of constantdepth circuits to enable longtime dynamic simulations.
Figure 3d and h compare the number of CNOT gates for each timestep in the constantdepth (red) and growing depth (green) circuits for 3 (dotted line), 4 (dashed lined), and 5spin (solid line) systems. Clearly the number of CNOT gates remains the same for all timesteps for our constantdepth circuits, but the number grows linearly with increasing numbers of timesteps for the growing depth circuits. Notice how the number of CNOT gates per timestep for the XY model circuits (3h) are approximately double the number for the TFIM circuits (3d), while our constantdepth circuits have the same CNOT count for both models.
Discussion and outlook
Standard Hamiltonian simulation algorithms produce circuits that grow in depth with increasing timestep count, which limits the number of timesteps that are feasible to simulate on nearterm quantum devices. The constantdepth circuits we have presented remove this limit when simulating a specific set of 1D Hamiltonian models, namely those which can be mapped to free fermionic models, including the TFIM and XY model. Simulations of these systems, therefore, can be broken into arbitrarily many timesteps, which allows for Trotter error to be made negligibly small and enables longtime dynamics to be more feasibly simulated on nearterm quantum devices. While simulations of freefermionic systems are known to be classically simulatable with resources that scale polynomially with system size (Valiant 2002; Terhal and DiVincenzo 2002), there are numerous ways in which the constantdepth can indirectly contribute to progressing dynamic simulations on materials on nearterm quantum computers. For example, our constantdepth circuits can enable preparation of nontrivial ground states of \(\mathcal {H_{CD}}\) models through adiabatic state preparation (ASP). Since our compressed circuits allow for arbitrarily many timesteps in a simulation, this allows arbitrarily slow (i.e., adiabatic) evolution under a timedependent Hamiltonian, which is the basis for ASP (AspuruGuzik et al. 2005; Barends et al. 2016). In this way, our circuits might be used as an initial state preparation subcircuit within a larger simulation circuit. Indeed, our compressed circuits may serve as subcircuits in any larger circuit that contains a component with timeevolution under one of the \(\mathcal {H_{CD}}\) Hamiltonians. Another example can be seen in Ref. Bassman et al. (2021), where a finite temperature state was prepared in an initial part of a simulation circuit and our compressed circuit was appended to time evolve the initial thermal state.
Other areas of utility for our compressed circuits include the benchmarking of new error mitigation or noise extrapolation techniques (Gustafson et al. 2019; Li and Benjamin 2017), as well as the benchmarking of the performance of quantum hardware in general. For these endeavours our constantdepth circuits can prove highly useful as they are feasible to run on nearterm hardware and are classically efficiently simulatable, which allows for the calculation of a ground truth for use in the benchmarking. Finally, future work might also explore if and how our constantdepth circuits can be adapted for approximations of various extensions of the Hamiltonians in \(\mathcal {H_{CD}}\), including twodimensional models, models with nextnearest neighbor or even longerrange interactions, or full Heisenberg interactions (i.e., coupling interactions along three axes). Indeed, matchgates have previously been studied for various twodimensional qubit topologies and for longerrange interactions (Brod and Galvão 2012; Brod and Childs 2014). Paired with incremental improvements in quantum hardware, the ability to extend our constantdepth circuits to more complex systems could pave the way to new discoveries in the behavior of quantum materials by enabling longtime dynamic simulations on quantum computers of systems relevant to scientific and technological problems.
Appendix A: quantum circuits for \(\mathcal {H}_{\mathcal {CD}}\) matchgates
Here, we describe the matchgates used for the various Hamiltonians in \(\mathcal {H_{CD}}\) in terms of their matrix representation, as well as their quantum circuit representation. In the following, we use θ_{i} to represent the free parameters of the circuit that must be set by the optimizer for a particular Hamiltonian and timestep. R_{x}(θ),R_{y}(θ), and R_{z}(θ) are rotation gates, which rotate the qubit around the x, y, zaxis, respectively, by an angle θ. Elements of the matchgate matrix representations in plain text are real numbers, while elements in colored, boldface text are complex numbers. A bar over a complex element denotes its complex conjugate.
Hamiltonian parameter subsets with: J _{x} J _{y} =J _{x} J _{z} =J _{y} J _{z} =0 and h _{β} =0
Matchgates used in constantdepth circuits for simulating \(H = j_{x} \sum _{i} \sigma _{i}^{x} \sigma _{i+1}^{x} \)
Matchgates used in constantdepth circuits for simulating \(H = j_{y} \sum _{i} \sigma _{i}^{y} \sigma _{i+1}^{y} \)
Matchgates used in constantdepth circuits for simulating \(H = j_{z} \sum _{i} \sigma _{i}^{z} \sigma _{i+1}^{z} \)
Hamiltonian parameter subsets with: J _{x} J _{y} =J _{x} J _{z} =J _{y} J _{z} =0 and h _{β} ≠0
G gates used in constantdepth circuits for simulating \(H = j_{x} \sum _{i} \sigma _{i}^{x} \sigma _{i+1}^{x} + h_{x}\sum _{i} \sigma _{i}^{x} \)
G gates used in constantdepth circuits for simulating \(H = j_{y} \sum _{i} \sigma _{i}^{y} \sigma _{i+1}^{y} + h_{y}\sum _{i} \sigma _{i}^{y} \)
Matchgates used in constantdepth circuits for simulating \(H = j_{z} \sum _{i} \sigma _{i}^{z} \sigma _{i+1}^{z} + h_{z}\sum _{i} \sigma _{i}^{z} \)
Hamiltonian parameter subsets with: J _{x} J _{y} J _{z} =0 and h _{β} =0
Matchgates used in constantdepth circuits for simulating \(H = \sum _{i} j_{x}\sigma _{i}^{x} \sigma _{i+1}^{x} + j_{y}\sigma _{i}^{y} \sigma _{i+1}^{y} \)
Matchgates used in constantdepth circuits for simulating \(H = \sum _{i} j_{x}\sigma _{i}^{x} \sigma _{i+1}^{x} + j_{z}\sigma _{i}^{z} \sigma _{i+1}^{z} \)
Matchgates used in constantdepth circuits for simulating \(H = \sum _{i} j_{y}\sigma _{i}^{y} \sigma _{i+1}^{y} + j_{z}\sigma _{i}^{z} \sigma _{i+1}^{z} \)
Hamiltonian parameter subsets with: J _{x} J _{y} J _{z} =0 and h _{β} ≠0
Matchgates used in constantdepth circuits for simulating \(H = j_{x}\sum _{i} \sigma _{i}^{x} \sigma _{i+1}^{x} + h_{z}\sum _{i} \sigma _{i}^{z}\) or \(H = j_{y}\sum _{i} \sigma _{i}^{y} \sigma _{i+1}^{y} + h_{z}\sum _{i} \sigma _{i}^{z}\) or \(H = \sum _{i} \{J_{x}\sigma _{i}^{x} \sigma _{i+1}^{x} + j_{y}\sigma _{i}^{y} \sigma _{i+1}^{y}\} + h_{z}\sum _{i} \sigma _{i}^{z}\)
G gates used in constantdepth circuits for simulating \(H = j_{x}\sum _{i} \sigma _{i}^{x} \sigma _{i+1}^{x} + h_{y}\sum _{i} \sigma _{i}^{y}\) or \(H = j_{z}\sum _{i} \sigma _{i}^{z} \sigma _{i+1}^{z} + h_{y}\sum _{i} \sigma _{i}^{y}\) or \(H = \sum _{i} \{J_{x}\sigma _{i}^{x} \sigma _{i+1}^{x} + j_{z}\sigma _{i}^{z} \sigma _{i+1}^{z}\} + h_{y}\sum _{i} \sigma _{i}^{y}\)
G gates used in constantdepth circuits for simulating \(H = j_{y}\sum _{i} \sigma _{i}^{y} \sigma _{i+1}^{y} + h_{x}\sum _{i} \sigma _{i}^{x}\) or \(H = j_{z}\sum _{i} \sigma _{i}^{z} \sigma _{i+1}^{z} + h_{x}\sum _{i} \sigma _{i}^{x}\) or \(H = \sum _{i} \{J_{y}\sigma _{i}^{y} \sigma _{i+1}^{y} + j_{z}\sigma _{i}^{z} \sigma _{i+1}^{z}\} + h_{x}\sum _{i} \sigma _{i}^{x}\)
Appendix B: Proof of Fig. 1a
Availability of data and material
Code to generate our constantdepth circuits can be found in the following GitHub repository: https://github.com/lebassman/Constant_Depth_Circuits.
Abbreviations
 1D:

OneDimensional
 NISQ:

Noisy IntermediateScale Quantum
 TFIM:

Transverse Field Ising Model
 ASP:

Adiabatic State Preparation
References
D. S. Abrams, S. Lloyd, Simulation of manybody Fermi systems on a universal quantum computer. Phys. Rev. Lett.79(13), 2586–2589 (1997). https://doi.org/10.1103/PhysRevLett.79.2586.
M. Acharyya, Nonequilibrium phase transition in the kinetic ising model: Is the transition point the maximum lossy point?Phys. Rev. E. 58(1), 179 (1998).
M. Acharyya, B. K. Chakrabarti, Response of ising systems to oscillating and pulsed fields: Hysteresis, ac, and pulse susceptibility. Phys. Rev. B. 52(9), 6550 (1995).
A. AspuruGuzik, A. D. Dutoi, P. J. Love, M. HeadGordon, Simulated quantum computation of molecular energies. Science. 309(5741), 1704–1707 (2005).
Y. Atia, D. Aharonov, Fastforwarding of Hamiltonians and exponentially precise measurements. Nat. Commun.8(1), 1572 (2017). https://doi.org/10.1038/s41467017016377.
R. Barends, A. Shabani, L. Lamata, J. Kelly, A. Mezzacapo, U. Las Heras, R. Babbush, A. G. Fowler, B. Campbell, Y. Chen, et al, Digitized adiabatic quantum computing with a superconducting circuit. Nature. 534(7606), 222–226 (2016).
L. Bassman, S. Gulania, C. Powers, R. Li, T. Linker, K. Liu, T. K. S. Kumar, R. K. Kalia, A. Nakano, P. Vashishta, Domainspecific compilers for dynamic simulations of quantum materials on quantum computers. Quantum Sci. Technol.6(1), 14007 (2020). https://doi.org/10.1088/20589565/abbea1.
L. Bassman, K. Klymko, N. M. Tubman, W. A. de Jong, Computing free energies with fluctuation relations on quantum computers (2021). https://arxiv.org/abs/2103.09846. Accessed 15 Dec 2021.
L. Bassman, K. Liu, A. Krishnamoorthy, T. Linker, Y. Geng, D. Shebib, S. Fukushima, F. Shimojo, R. K. Kalia, A. Nakano, et al, Towards simulation of the dynamics of materials on quantum computers. Phys. Rev. B. 101(18), 184305 (2020). https://doi.org/10.1103/PhysRevB.101.184305.
L. Bassman, M. Urbanek, M. Metcalf, J. Carter, A. F. Kemper, W. A. de Jong, Simulating Quantum Materials with Digital Quantum Computers. Quantum Sci. Technol.6:, 043002 (2021).
L Bassman, R Van Beeumen, E Younis, E Smith, Constant Depth Circuits for Dynamic Simulations of Materials on Quantum Computers (2021). https://github.com/lebassman/Constant_Depth_Circuits. Accessed 15 Dec 2021.
Berkeley Quantum Synthesis Toolkit (2021). https://bqskit.lbl.gov. Accessed 15 Dec 2021.
D. W. Berry, G. Ahokas, R. Cleve, B. C. Sanders, Efficient quantum algorithms for simulating sparse Hamiltonians. Comm. Math. Phys.270(2), 359–371 (2007). https://doi.org/10.1007/s002200060150x.
R. Blinc, B. žekš, J. Sampaio, A. Pires, F. S. Barreto, Ising model in a transverse tunneling field and protonlattice interaction in Hbonded ferroelectrics. Phys. Rev. B. 20(5), 1991 (1979).
A. Botea, A. Kishimoto, R. Marinescu, in Proceedings of the 11th International Symposium on Combinatorial Search (SoCS 2018). On the complexity of quantum circuit compilation (AAAI PressPalo Alto, 2018), pp. 138–142.
D. J. Brod, A. M. Childs, The computational power of matchgates and the XY interaction on arbitrary graphs. Quantum Inf. Comput.14(1112), 901–916 (2014). https://doi.org/10.26421/QIC14.1112.
D. J. Brod, E. F. Galvão, Geometries for universal quantum computation with matchgates. Phys. Rev. A. 86(5), 52307 (2012). https://doi.org/10.1103/PhysRevA.86.052307.
E. Campbell, Random compiler for fast hamiltonian simulation. Phys. Rev. Lett.123(7), 070503 (2019).
Y. H. Chen, A. Kalev, I. Hen, A quantum algorithm for timedependent hamiltonian simulation by permutation expansion (2021). https://arxiv.org/abs/2103.15334. Accessed 15 Dec 2021.
H. Cheraghi, S. Mahdavifar, probing the possibilities of ergodicity in the 1d spin1/2 xy chain with quench dynamics. Sci. Rep.10(1), 1–10 (2020).
A. M. Childs, R. Kothari, Limitations on the simulation of nonsparse Hamiltonians. Quantum Inf. Comput.10(78), 669–684 (2010). https://doi.org/10.26421/QIC10.787.
A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, Y. Su, Toward the first quantum simulation with quantum speedup. Proc. Natl. Acad. Sci.115(38), 9456–9461 (2018).
A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, S. Zhu, Theory of trotter error with commutator scaling. Phys. Rev. X. 11(1), 011020 (2021).
A. M. Childs, N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations (2012). https://arxiv.org/abs/1202.5822. Accessed 15 Dec 21.
L. Cincio, K. Rudinger, M. Sarovar, P. J. Coles, Machine learning of noiseresilient quantum circuits. PRX Quantum. 2:, 010324 (2020).
C. Cîrstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, A. Sornborger, Variational fast forwarding for quantum simulation beyond the coherence time. NPJ Quantum Inf.6(1), 82 (2020). https://doi.org/10.1038/s41534020003020.
A. De Vos, S. De Baerdemacker, BlockZXZ synthesis of an arbitrary quantum circuit. Phys. Rev. A.94(5), 52317 (2016). https://doi.org/10.1103/PhysRevA.94.052317.
R. P. Feynman, Simulating physics with computers. Int. J. Theor. Phys.21(6), 467–488 (1982). https://doi.org/10.1007/BF02650179.
F. GómezRuiz, J. MendozaArenas, F. Rodríguez, C. Tejedor, L. Quiroga, Quantum phase transitions detected by a local probe using time correlations and violations of leggettgarg inequalities. Phys. Rev. B. 93(3), 035441 (2016).
E. Granato, Critical behavior of a onedimensional frustrated quantum xy model. Phys. Rev. B. 45(5), 2557 (1992).
S. Gu, R. D. Somma, B. Şahinoğlu, Fastforwarding quantum evolution (2021). https://arxiv.org/abs/2105.07304. Accessed 15 Dec 21.
E. Gustafson, P. Dreher, Z. Hang, Y. Meurice, Benchmarking quantum computers for realtime evolution of a (1+1) field theory with error mitigation. arXiv preprint arXiv:1910.09478 (2019).
E. Gustafson, Y. Meurice, J. UnmuthYockey, Quantum simulation of scattering in the quantum ising model. Phys. Rev. D. 99(9), 094503 (2019).
D. Herr, F. Nori, S. J. Devitt, Optimization of lattice surgery is NPhard. NPJ Quantum Inf.3(1), 35 (2017). https://doi.org/10.1038/s4153401700351.
R. Iten, R. Colbeck, I. Kukuljan, J. Home, M. Christandl, Quantum circuits for isometries. Phys. Rev. A. 93(3), 032318 (2016). https://doi.org/10.1103/PhysRevA.93.032318.
S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sornborger, P. J. Coles, Quantumassisted quantum compiling. Quantum. 3:, 140 (2019). https://doi.org/10.22331/q20190513140.
I. D. Kivlichan, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, W. Sun, Z. Jiang, N. Rubin, A. Fowler, A AspuruGuzik, et al, Improved faulttolerant quantum simulation of condensedphase correlated electrons via trotterization. Quantum. 4:, 296 (2020).
H. Lamm, S. Lawrence, Simulation of nonequilibrium dynamics on a quantum computer. Phys. Rev. Lett.121(17), 170501 (2018).
Y. Li, S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X. 7(2), 021050 (2017).
S. Lloyd, Universal quantum simulators. Science. 273(5278), 1073–1078 (1996). https://doi.org/10.1126/science.273.5278.1073.
E. A. Martinez, T. Monz, D. Nigg, P. Schindler, R. Blatt, Compiling quantum algorithms for architectures with multiqubit gates. New J. Phys.18(6), 063029 (2016). https://doi.org/10.1088/13672630/18/6/063029.
M. Möttönen, J. J. Vartiainen, V. Bergholm, M. M. Salomaa, Quantum circuits for general multiqubit gates. Phys. Rev. Lett.93(13), 130502 (2004). https://doi.org/10.1103/PhysRevLett.93.130502.
P. Murali, J. M. Baker, A. JavadiAbhari, F. T. Chong, M. Martonosi, in Proceedings of the TwentyFourth International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS ’19). NoiseAdaptive Compiler Mappings for Noisy IntermediateScale Quantum Computers (ACM, 2019), pp. 1015–1029. https://doi.org/10.1145/3297858.3304075.
C. Neill, T. McCourt, X. Mi, Z. Jiang, M. Niu, W. Mruczkiewicz, I. Aleiner, F. Arute, K. Arya, J. Atalaya, et al, Accurately computing the electronic properties of a quantum ring. Nature. 594(7864), 508–512 (2021).
D. Poulin, A. Qarry, R. Somma, F. Verstraete, Quantum simulation of timedependent Hamiltonians and the convenient illusion of Hilbert space. Phys. Rev. Lett.106(17), 170501 (2011). https://doi.org/10.1103/PhysRevLett.106.170501.
J. Preskill, Quantum Computing in the NISQ era and beyond. Quantum. 2:, 79 (2018). https://doi.org/10.22331/q2018080679.
S. Sides, P. Rikvold, M. Novotny, Kinetic ising model in an oscillating field: Finitesize scaling at the dynamic phase transition. Phys. Rev. Lett.81(4), 834 (1998).
A. Smith, M. S. Kim, F. Pollmann, J. Knolle, Simulating quantum manybody dynamics on a current digital quantum computer. NPJ Quantum Inf.5(1), 106 (2019). https://doi.org/10.1038/s4153401902170.
S. N. Sun, M. Motta, R. N. Tazhigulov, A. T. Tan, G. K. L. Chan, A. J. Minnich, Quantum computation of finitetemperature static and dynamical properties of spin systems using quantum imaginary time evolution. PRX Quantum. 2(1), 010317 (2021).
S. Suzuki, J. Inoue, B. K. Chakrabarti, Quantum Ising Phases and Transitions in Transverse Ising Models (Springer Berlin Heidelberg, Germany, 2012). https://books.google.com/books?id=y1S5BQAAQBAJ.
B. M. Terhal, D. P. DiVincenzo, Classical simulation of noninteractingfermion quantum circuits. Phys. Rev. A. 65(3), 032325 (2002).
T. Tomé, M. J. de Oliveira, Dynamic phase transition in the kinetic ising model under a timedependent oscillating field. Phys. Rev. A. 41(8), 4251 (1990).
M. C. Tran, S. K. Chu, Y. Su, A. M. Childs, A. V. Gorshkov, Destructive error interference in productformula lattice simulation. Phys. Rev. Lett.124(22), 220502 (2020).
H. F. Trotter, On the product of semigroups of operators. Proc. Amer. Math. Soc.10(4), 545–551 (1959). https://doi.org/10.1112/jlms/s138.1.66.
L. G. Valiant, Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput.31(4), 1229–1254 (2002). https://doi.org/10.1137/S0097539700377025.
G. Vidal, C. M. Dawson, Universal quantum circuit for twoqubit transformations with three controlledNOT gates. Phys. Rev. A. 69(1), 10301 (2004). https://doi.org/10.1103/PhysRevA.69.010301.
N. Wiebe, D. W. Berry, P. Høyer, B. C. Sanders, Simulating quantum dynamics on a quantum computer. J. Phys. A Math. Theor.44(44), 445308 (2011). https://doi.org/10.1088/17518113/44/44/445308.
W. Wu, B. Ellman, T. Rosenbaum, G. Aeppli, D. Reich, From classical to quantum glass. Phys. Rev. Lett.67(15), 2076 (1991).
X. Yang, G. Zhang, Z. Song, Phase transition in phase transition lines of quantum xy model. J. Phys. Condens. Matter. 31(24), 245401 (2019).
K. YeterAydeniz, G. Siopsis, R. C. Pooser, Scattering in the ising model with the quantum lanczos algorithm. New J. Phys.23(4), 043033 (2021). https://doi.org/10.1088/13672630/abe63d.
E. Younis, K. Sen, K. Yelick, C. Iancu, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE). QFAST: Conflating Search and Numerical Optimization for Scalable Quantum Circuit Synthesis, (2021), pp. 232–243. https://doi.org/10.1109/QCE52317.2021.00041.
C. Zalka, Simulating quantum systems on a quantum computer. Proc. R. Soc. A. 454(1969), 313–322 (1998). https://doi.org/10.1098/rspa.1998.0162.
D. Zhu, S. Johri, N. M. Linke, K. Landsman, C. H. Alderete, N. H. Nguyen, A. Matsuura, T. Hsieh, C. Monroe, Generation of thermofield double states and critical ground states with a quantum computer. Proc. Natl. Acad. Sci.117(41), 25402–25406 (2020).
A. Zhukov, S. Remizov, W. Pogosov, Y. E. Lozovik, Algorithmic simulation of farfromequilibrium dynamics using quantum computer. Quantum Inf. Process.17(9), 223 (2018).
Acknowledgements
We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.
Funding
This work was supported by the U.S. Department of Energy (DOE) under Contract No. DEAC0205CH11231, through the Office of Advanced Scientific Computing Research Accelerated Research for Quantum Computing Program (LB, RVB, EY, CI, and WAdJ) and the Advanced Quantum Testbed (ES). This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract No. DEAC0500OR22725.
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LB, RVB, and WAdJ designed the research. All authors aided in the development of code for generating the constant depth circuits. RVB developed the theoretical framework. LB performed the experimental simulations. LB and RVB prepared the manuscript. All authors read and approved the final manuscript.
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Bassman, L., Van Beeumen, R., Younis, E. et al. Constantdepth circuits for dynamic simulations of materials on quantum computers. Mater Theory 6, 13 (2022). https://doi.org/10.1186/s4131302200043x
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DOI: https://doi.org/10.1186/s4131302200043x
Keywords
 Quantum simulation
 Quantum computation
 Quantum circuit synthesis
 Materials simulation
 Dynamic simulation