Trotter Formula Research Articles

Operator Relationship between Conventional Coupled Cluster and Unitary Coupled Cluster

The chemistry community has long sought the exact relationship between the conventional and the unitary coupled cluster ansatz for a single-reference system, especially given the interest in performing quantum chemistry on quantum computers. In this work, we show how one can use the operator manipulations given by the exponential disentangling identity and the Hadamard lemma to relate the factorized form of the unitary coupled-cluster approximation to a factorized form of the conventional coupled cluster approximation (the factorized form is required, because some amplitudes are operator-valued and do not commute with other terms). By employing the Trotter product formula, one can then relate the factorized form to the standard form of the unitary coupled cluster ansatz. The operator dependence of the factorized form of the coupled cluster approximation can also be removed at the expense of requiring even more higher-rank operators, finally yielding the conventional coupled cluster. The algebraic manipulations of this approach are daunting to carry out by hand, but can be automated on a computer for small enough systems.

Flexibility of the factorized form of the unitary coupled cluster Ansatz.

The factorized form of the unitary coupled cluster Ansatz is a popular state preparation Ansatz for electronic structure calculations of molecules on quantum computers. It is often viewed as an approximation (based on the Trotter product formula) for the conventional unitary coupled cluster operator. In this work, we show that the factorized form is quite flexible, allowing one to range from a conventional configuration interaction, to conventional unitary coupled cluster, to efficient approximations that lie in between these two. The variational minimization of the energy often allows simpler factorized unitary coupled cluster approximations to achieve high accuracy, even if they do not accurately approximate the Trotter product formula. This is similar to how quantum approximate optimization algorithms can achieve high accuracy with a small number of levels.

Dynamic Analysis of Software Systems with Aperiodic Impulse Rejuvenation

This paper aims to obtain the dynamical solution and instantaneous availability of software systems with aperiodic impulse rejuvenation. Firstly, we formulate the generic system with a group of coupled impulsive differential equations and transform it into an abstract Cauchy problem. Then we adopt a difference scheme and establish the convergence of this scheme by applying the Trotter–Kato theorem to obtain the system’s dynamical solution. Moreover, the instantaneous availability as an important evaluation index for software systems is derived, and its range is also estimated. At last, numerical examples are shown to illustrate the validity of theoretical results.

Trotter–Kato product formula in symmetric F-normed ideals

We prove the Trotter–Kato product formula for arbitrary symmetric F-normed ideals which are closed with respect to logarithmic submajorization. This class of ideals include all symmetric quasi-Banach ideals.

Quantum Computer-Aided Design: Digital Quantum Simulation of Quantum Processors

With the increasing size of quantum processors, sub-modules that constitute the processor hardware will become too large to accurately simulate on a classical computer. Therefore, one would soon have to fabricate and test each new design primitive and parameter choice in time-consuming coordination between design, fabrication, and experimental validation. Here we show how one can design and test the performance of next-generation quantum hardware -- by using existing quantum computers. Focusing on superconducting transmon processors as a prominent hardware platform, we compute the static and dynamic properties of individual and coupled transmons. We show how the energy spectra of transmons can be obtained by variational hybrid quantum-classical algorithms that are well-suited for near-term noisy quantum computers. In addition, single- and two-qubit gate simulations are demonstrated via Suzuki-Trotter decomposition. Our methods pave a promising way towards designing candidate quantum processors when the demands of calculating sub-module properties exceed the capabilities of classical computing resources.

Finite-temperature symmetric tensor network for spin-1/2 Heisenberg antiferromagnets on the square lattice

Within the tensor network framework, the (positive) thermal density operator can be approximated by a double layer of infinite Projected Entangled Pair Operator (iPEPO) coupled via ancilla degrees of freedom. To investigate the thermal properties of the spin-1/2 Heisenberg model on the square lattice, we introduce a family of fully spin-$SU(2)$ and lattice-$C_{4v}$ symmetric on-site tensors (of bond dimensions $D=4$ or $D=7$) and a plaquette-based Trotter-Suzuki decomposition of the imaginary-time evolution operator. A variational optimization is performed on the plaquettes, using a full (for $D=4$) or simple (for $D=7$) environment obtained from the single-site Corner Transfer Matrix Renormalization Group fixed point. The method is benchmarked by a comparison to quantum Monte Carlo in the thermodynamic limit. Although the iPEPO spin correlation length starts to deviate from the exact exponential growth for inverse-temperature $\beta \gtrsim 2$, the behavior of various observables turns out to be quite accurate once plotted w.r.t the inverse correlation length. We also find that a direct $T=0$ variational energy optimization provides results in full agreement with the $\beta\rightarrow\infty$ limit of finite-temperature data, hence validating the imaginary-time evolution procedure. Extension of the method to frustrated models is described and preliminary results are shown.

Persistency of linear programming relaxations for the stable set problem

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.

Transient analysis of a single server queueing system with infinite buffer

As a typical single server queueing system, computer integrated manufacturing system (CIMS) has been widely used in the field of intelligent manufacturing. However, how to derive its instantaneous index is still an important issue. This paper investigates the transient behavior of the CIMS with spectral method. By constructing an asymptotic system and analyzing the spectral distribution, we derive the explicit transient solution of the asymptotic system. Trotter–Kato theorem is used to prove that the transient solution of the CIMS is just the limitation of explicit transient solution of the asymptotic system. At the end of the paper, numerical examples are shown to illustrate the effectiveness of the proposed approximation.

Toward Quantum Computing for High-Energy Excited States in Molecular Systems: Quantum Phase Estimations of Core-Level States.

This paper explores the utility of the quantum phase estimation (QPE) algorithm in calculating high-energy excited states characterized by the promotion of electrons occupying core-level shells. These states have been intensively studied over the last few decades, especially in supporting the experimental effort at light sources. Results obtained with QPE are compared with various high-accuracy many-body techniques developed to describe core-level states. The feasibility of the quantum phase estimator in identifying classes of challenging shake-up states characterized by the presence of higher-order excitation effects is discussed. We also demonstrate the utility of the QPE algorithm in targeting excitations from specific centers in a molecule. Lastly, we discuss how the lowest-order Trotter formula can be applied to reducing the complexity of the ansatz without affecting the error.

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

ABSTRACTWe propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.

Existence, uniqueness and an ODE approach to the L1-Karcher mean

We extend the domain of the Karcher mean Λ of positive operators on a Hilbert space to L1-Borel probability measures on the cone of positive operators equipped with the Thompson part metric. We establish existence and uniqueness of Λ as the solution of the Karcher equation and develop a nonlinear ODE theory for the relative operator entropy in the spirit of Crandall-Liggett, such that the solutions of the Karcher equation are stationary solutions of the ODE, and all generated solution curves enjoy the exponential contraction estimate. This is possible despite the facts that the Thompson metric is non-Euclidean, non-differentiable, non-commutative as a metric space as well as non-separable, and the positive cone is non-locally compact as a manifold. As further applications of the ODE approach, we prove the norm convergence conjecture of the power means of positive operators to the Karcher mean, and a Trotter-Kato product formula for the nonlinear semigroups explicitly expressed by compositions of two-variable geometric means. This can be regarded as a nonlinear continuous-time version of the law of large numbers.

Towards a quantum Monte Carlo for lattice systems

In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.

Quantum Algorithms for Simulating the Lattice Schwinger Model

The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model onN/2physical sites with coupling constantx−1/2and electric field cutoffx−1/2Λcan be simulated on a quantum computer for time2xTusing a number ofT-gates or CNOTs inO~(N3/2T3/2xΛ)for fixed operator error. This scaling with the truncationΛis better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable–the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested.

On Grids in Point-Line Arrangements in the Plane

The famous Szemerédi–Trotter theorem states that any arrangement of n points and n lines in the plane determines $$O(n^{4/3})$$ incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let $$\mathcal {L}_a$$ and $$\mathcal {L}_b$$ be two sets of t lines in the plane and let $$P=\{\ell _a \cap \ell _b : \ell _a \in \mathcal {L}_a, \,\ell _b \in \mathcal {L}_b\}$$ be the set of intersection points between $$\mathcal {L}_a$$ and $$\mathcal {L}_b$$ . We say that $$(P, \mathcal {L}_a \cup \mathcal {L}_b)$$ forms a natural $$t\times t$$ grid if $$|P| =t^2$$ , and $${\text {conv}}P$$ does not contain the intersection point of some two lines in $$\mathcal {L}_a$$ and does not contain the intersection point of some two lines in $$\mathcal {L}_b$$ . For fixed $$t > 1$$ , we show that any arrangement of n points and n lines in the plane that does not contain a natural $$t\times t$$ grid determines $$O(n^{{4}/{3}- \varepsilon })$$ incidences, where $$\varepsilon = \varepsilon (t)>0$$ . We also provide a construction of n points and n lines in the plane that does not contain a natural $$2 \times 2$$ grid and determines at least $$\Omega ({n^{1+{1}/{14}}})$$ incidences.

Sum–product phenomena for planar hypercomplex numbers

We study the sum–product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum–product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of max{|A+A|,|AA|} is neither close to |A| nor to |A|2.To obtain our main sum–product bound, we extend Elekes’ sum–product technique that relies on point–line incidences. Our extension is significantly more involved than the original proof, and in some sense runs the original technique a few times in a bootstrapping manner. We also study point–line incidences in the dual plane and in the double plane, developing analogs of the Szemerédi–Trotter theorem. As in the case of the sum–product problem, it turns out that the dual and double variants behave differently than the complex and real ones.

Synthetic mean-field interactions in photonic lattices

Photonic lattices are usually considered to be limited by their lack of methods to include interactions. We address this issue by introducing mean-field interactions through optical components which are external to the photonic lattice. The proposed technique to realise mean-field interacting photonic lattices relies on a Suzuki-Trotter decomposition of the unitary evolution for the full Hamiltonian. The technique realises the dynamics in an analogous way to that of a step-wise numerical implementation of quantum dynamics, in the spirit of digital quantum simulation. It is a very versatile technique which allows for the emulation of interactions that do not only depend on inter-particle separations or do not decay with particle separation. We detail the proposed experimental scheme and consider two examples of interacting phenomena, self-trapping and the decay of Bloch oscillations, that are observable with the proposed technique.Graphical abstract

Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms

Lie–Trotter–Suzuki decompositions are an efficient way to approximate operator exponentials exp(tH) when H is a sum of n (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. We provide optimized decompositions up to order t6. The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For n=2 terms, several of the optima we find are close to those in McLachlan (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.

Cutting lemma and Zarankiewicz’s problem in distal structures

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in o-minimal expansions of fields. Using it, we generalize the results in Fox et al. (J Eur Math Soc 19(6):1785–1810, 2017 ) on the semialgebraic planar Zarankiewicz problem to arbitrary o-minimal structures, in particular obtaining an o-minimal generalization of the Szemeredi–Trotter theorem.

Renormalized Circuit Complexity.

We propose a modification to Nielsen's circuit complexity for Hamiltonian simulation using the Suzuki-Trotter (ST) method, which provides a network like structure for the quantum circuit. This leads to an optimized gate counting linear in the geodesic distance and spatial volume, unlike in the original proposal. The optimized ST iteration order is correlated with the error tolerance and plays the role of an anti-de Sitter radial coordinate. The density of gates is shown to be monotonic with the tolerance and a holographic interpretation using path-integral optimization is given.

Optimisation-based representations for branching processes

It is shown that a certain functional of a branching process has representations in terms of both a maximisation problem and a minimisation problem. A consequence of these representations is that upper and lower bounds on the functional can be found easily, yielding a non-asymptotic Trotter product formula. As an application, the speed of the right-most particle of a branching Levy process is calculated.