Hamiltonian Simulation Research Articles

A quantum algorithm for track reconstruction in the LHCb vertex detector

High-energy physics is facing increasingly demanding computational challenges in real-time event reconstruction for the near-future high-luminosity era. Using the LHCb vertex detector as a use case, we explore a new algorithm for particle track reconstruction based on the minimisation of an Ising-like Hamiltonian with a linear algebra approach. The use of a classical matrix inversion technique results in tracking performance similar to the current state-of-the-art but with worse scaling complexity in time. To solve this problem, we also present an implementation as a quantum algorithm, using the Harrow-Hassadim-Lloyd (HHL) algorithm: this approach can potentially provide an exponential speedup as a function of the number of input hits over its classical counterpart, in spite of limitations due to the well-known HHL Hamiltonian simulation and readout problems. The findings presented in this paper shed light on the potential of leveraging quantum computing for real-time particle track reconstruction in high-energy physics.

Linear Combination of Hamiltonian Simulation for Nonunitary Dynamics with Optimal State Preparation Cost.

We propose a simple method for simulating a general class of nonunitary dynamics as a linear combination of Hamiltonian simulation (LCHS) problems. LCHS does not rely on converting the problem into a dilated linear system problem or on the spectral mapping theorem. The latter is the mathematical foundation of many quantum algorithms for solving a wide variety of tasks involving nonunitary processes, such as the quantum singular value transformation. The LCHS method can achieve optimal cost in terms of state preparation. We also demonstrate an application for open quantum dynamics simulation using the complex absorbing potential method with near-optimal dependence on all parameters.

Problem-specific classical optimization of Hamiltonian simulation

Problem-specific classical optimization of Hamiltonian simulation

Accelerating quantum optimal control of multi-qubit systems with symmetry-based Hamiltonian transformations

We present a novel, computationally efficient approach to accelerate quantum optimal control calculations of large multi-qubit systems used in a variety of quantum computing applications. By leveraging the intrinsic symmetry of finite groups, the Hilbert space can be decomposed and the Hamiltonians block diagonalized to enable extremely fast quantum optimal control calculations. Our approach reduces the Hamiltonian size of an n-qubit system from 2n×2n to O(n×n) or O((2n/n)×(2n/n)) under Sn or Dn symmetry, respectively. Most importantly, this approach reduces the computational runtime of qubit optimal control calculations by orders of magnitude while maintaining the same accuracy as the conventional method. As prospective applications, we show that (1) symmetry-protected subspaces can be potential platforms for quantum error suppression and simulation of other quantum Hamiltonians and (2) Lie–Trotter–Suzuki decomposition approaches can generalize our method to a general variety of multi-qubit systems.

Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis

The Quantum State Preparation problem aims to prepare an n-qubit quantum state |ψv=k=02n-1vk|k from the initial state |0n, for a given unit vector v=(v0,v1,v2,v2n-1)TC2n with ||v||2=1. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth complexity remains open when ancillary qubits are available. In this paper, we study quantum circuits when there are m ancillary qubits available. We construct, for any m, circuits that can prepare |ψvin depth Õ(2nm+n+n) and size O(2n), achieving the optimal value for both measures simultaneously. These results also imply a depth complexity of (4nm+n) for quantum circuits implementing a general n-qubit unitary for any m≤O(2n/n) number of ancillary qubits. This resolves the depth complexity for circuits without ancillary qubits. And for circuits with exponentially many ancillary qubits, our result quadratically improves the currently best upper bound of O(4n) to ˜(2n). Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide parameter regime. The results can be viewed as (optimal) time-space trade-off bounds, which is not only theoretically interesting, but also practically relevant in the current trend that the number of qubits starts to take off, by showing a way to use a large number of qubits to compensate the short qubit lifetime.

Realization of quantum signal processing on a noisy quantum computer

Quantum signal processing (QSP) is a powerful toolbox for the design of quantum algorithms and can lead to asymptotically optimal computational costs. Its realization on noisy quantum computers without fault tolerance, however, is challenging because it requires a deep quantum circuit in general. We propose a strategy to run an entire QSP protocol on noisy quantum hardware by carefully reducing overhead costs at each step. To illustrate the approach, we consider the application of Hamiltonian simulation for which QSP implements a polynomial approximation of the time evolution operator. We test the protocol by running the algorithm on the Quantinuum H1-1 trapped-ion quantum computer powered by Honeywell. In particular, we compute the time dependence of bipartite entanglement entropies for Ising spin chains and find good agreements with exact numerical simulations. To make the best use of the device, we determine optimal experimental parameters by using a simplified error model for the hardware and numerically studying the trade-off between Hamiltonian simulation time, polynomial degree, and total accuracy. Our results are the first step in the experimental realization of QSP-based quantum algorithms.

Hamiltonian simulation in the Pauli basis of multi-qubit clusters for condensed matter physics

We propose an efficient method for Hamiltonian simulation of multi-qubit quantum systems with special types of interaction. In our approach, the Hamiltonian of a \(n\)-qubit system should be represented as a linear combination of the standard Pauli basis operators, and then decomposed into a sum of partial Hamiltonians, which are, in general, not Pauli operators and satisfy some anticommutation relations. For three types of Hamiltonians, which are invariant with respect to permutations of qubits, the effectiveness of the main algorithm in the three-qubit cluster model is shown by calculating the operator exponentials for these Hamiltonians in an explicit analytical form. We also calculate the density operator, partition function, entropy, and free energy of the cluster weakly coupled to a thermal environment. In our model, the cluster is in the Gibbs state in the temperature interval \(0.1\!-\!2\:\!\text{K}\), which corresponds to the operating range of modern quantum processors. It follows from our analysis that the thermodynamic properties of such systems strongly depend on the type of internal interaction of qubits in the cluster.

Efficient and accurate calculation of proline cis/trans isomerization free energies from Hamiltonian replica exchange molecular dynamics simulations

Proline cis/trans isomerization plays an important role in many biological processes but occurs on time scales not accessible to brute-force molecular dynamics (MD) simulations. We have designed a new Hamiltonian replica exchange scheme, ω-bias potential replica exchange molecular dynamics (ωBP-REMD), to efficiently and accurately calculate proline cis/trans isomerization free energies. ωBP-REMD is applied to various proline-containing tripeptides and a biologically important proline residue in the N2-domain of the gene-3-protein of phage fd in the wildtype and mutant variants of the protein. Excellent cis/trans transition rates are obtained. Reweighting of the sampled probability distribution along the peptide bond dihedral angle allows construction of the corresponding free-energy profile and calculation of the cis/trans isomerization free energy with high statistical precision. Very good agreement with experimental data is obtained. ωBP-REMD outperforms standard umbrella sampling in terms of convergence and agreement with experiment and strongly reduces perturbation of the local structure near the proline residue.

Optimal and nearly optimal simulation of multiperiodic time-dependent Hamiltonians

Optimal and nearly optimal simulation of multiperiodic time-dependent Hamiltonians

Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory

Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space mathcal {H}. Here we give a prescription for gauge links and canonical momenta of an SU(2) gauge theory, such that the matrix representation of the former is diagonal in mathcal {H}. This is achieved by discretising the sphere S_3 isomorphic to SU(2) and the corresponding directional derivatives. We show that the fundamental commutation relations are fulfilled up to discretisation artefacts. Moreover, we directly construct the Casimir operator corresponding to the Laplace–Beltrami operator on S_3 and show that the spectrum of the free theory is reproduced again up to discretisation effects. Qualitatively, these results do not depend on the specific discretisation of SU(2), but the actual convergence rates do.

Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation

Simulating the time-evolution of a Hamiltonian is one of the most promising applications of quantum computers. Multi-Product Formulas (MPFs) are well suited to replace standard product formulas since they scale better with respect to time and approximation errors. Hamiltonian simulation with MPFs was first proposed in a fully quantum setting using a linear combination of unitaries. Here, we analyze and demonstrate a hybrid quantum-classical approach to MPFs that classically combines expectation values evaluated with a quantum computer. This has the same approximation bounds as the fully quantum MPFs, but, in contrast, requires no additional qubits, no controlled operations, and is not probabilistic. We show how to design MPFs that do not amplify the hardware and sampling errors, and demonstrate their performance. In particular, we illustrate the potential of our work by theoretically analyzing the benefits when applied to a classically intractable spin-boson model, and by computing the dynamics of the transverse field Ising model using a classical simulator as well as quantum hardware. We observe an error reduction of up to an order of magnitude when compared to a product formula approach by suppressing hardware noise with Pauli Twirling, pulse efficient transpilation, and a novel zero-noise extrapolation based on scaled cross-resonance pulses. The MPF methodology reduces the circuit depth and may therefore represent an important step towards quantum advantage for Hamiltonian simulation on noisy hardware.

Programmable Quantum Simulations of Bosonic Systems with Trapped Ions.

Trapped atomic ion crystals are a leading platform for quantum simulations of spin systems, with programmable and long-range spin-spin interactions mediated by excitations of phonons in the crystal. We describe a complementary approach for quantum simulations of bosonic systems using phonons in trapped-ion crystals, here mediated by excitations of the trapped-ion spins. The scheme enables a high degree of programability across a dense graph of bosonic couplings, utilizing long-lived collective phonon modes in a trapped-ion chain. As such, it is well suited for tackling hard problems such as boson sampling and simulations of long-range bosonic and spin-boson Hamiltonians.

Quantum algorithm for the linear Vlasov equation with collisions

The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. This representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.

Classically optimized Hamiltonian simulation

Hamiltonian simulation is a promising application for quantum computers to achieve a quantum advantage. We present classical algorithms based on tensor network methods to optimize quantum circuits for this task. We show that, compared to Trotter product formulas, the classically optimized circuits can be orders of magnitude more accurate and significantly extend the total simulation time.

Reducing molecular electronic Hamiltonian simulation cost for linear combination of unitaries approaches

We consider different linear combination of unitaries (LCU) decompositions for molecular electronic structure Hamiltonians. Using these LCU decompositions for Hamiltonian simulation on a quantum computer, the main figure of merit is the 1-norm of their coefficients, which is associated with the quantum circuit complexity. It is derived that the lowest possible LCU 1-norm for a given Hamiltonian is half of its spectral range. This lowest norm decomposition is practically unattainable for general Hamiltonians; therefore, multiple practical techniques to generate LCU decompositions are proposed and assessed. A technique using symmetries to reduce the 1-norm further is also introduced. In addition to considering LCU in the Schrödinger picture, we extend it to the interaction picture, which substantially further reduces the 1-norm.

Measurement-based quantum simulation of Abelian lattice gauge theories

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and their quantum simulation is expected to become a major application of quantum computers in the future. In this work, for an Abelian lattice gauge theory in dd spacetime dimensions, we define an entangled resource state (generalized cluster state) that reflects the spacetime structure of the gauge theory. We show that sequential single-qubit measurements with the bases adapted according to the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the gauge theory on the boundary. Our construction includes the (2+1)(2+1)-dimensional Abelian lattice gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the simulation of Wegner’s lattice models M_{(d,n)}M(d,n) that involve higher-form Abelian gauge fields. We demonstrate that the generalized cluster state has a symmetry-protected topological order with respect to generalized global symmetries that are related to the symmetries of the simulated gauge theories on the boundary.Our procedure can be generalized to the simulation of Kitaev’s Majorana chain on a fermionic resource state. We also study the imaginary-time quantum simulation with two-qubit measurements and post-selections, and a classical-quantum correspondence, where the statistical partition function of the model M_{(d,n)}M(d,n) is written as the overlap between the product of two-qubit measurement bases and the wave function of the generalized cluster state.

Co-segregation of Y and Zr in W-Cr-Y-Zr alloys: First-principles modeling at finite temperature and application to SMART materials

Spinodal phase separation in SMART (Self-passivating Metal Alloys with Reduced Thermo-oxidation) materials based on binary W-Cr with alloying elements Y and Zr is systematically investigated by a combination of Density Functional Theory with Cluster Expansion Hamiltonian and large-scale Monte Carlo simulations with thermodynamic integration. Comparing alloys of Zr with those of Y in ternary compositions with W and Cr, it is shown that there is a significant difference in the average short-range order between W and Zr, which changes from positive to negative from 500 K to higher temperatures in W70Cr29Zr1 alloys compared with the positive SRO (Short-Range Order) between W and Y in W70Cr29Y1 alloys at all temperatures. This change, however, does not affect the segregation behavior between W and Cr in the whole range of temperature between these alloys. Importantly, it is found that co-segregation of Y and Zr due to the negative SRO between them, in W70Cr29Y0.5Zr0.5 and W70Cr28Y1Zr1 considered alloys, increases the W-Cr positive SRO closer to the binary W70Cr30. Our modeling results reveal a significant impact on spinodal phase segregation of W and Cr in designing SMART materials for DEMO devices of future fusion power plants. Findings from recent experimental studies validate the modeling results, particularly the high-temperature behavior of the W-Cr-Zr alloys and the microstructure evolution of the W-Cr-Y-Zr alloys. The results provide a fundamental understanding of the radiation resistance phenomena observed in recent experiments on neutron-irradiated self-passivating alloys.

A Complexity-Efficient Quantum Architecture and Simulation for Eigen Spectrum Estimation of Vandermonde System in a Large Antenna Array

Vandermonde matrix finds a wide range of applications in the digital signal processing domain including antenna array processing. Eigenspectrum estimation remains a challenge in such a system when the matrix dimension increases. This precise problem can be well addressed in a quantum simulation framework based on quantum Hamiltonian simulation (QHS). However, the structural exploitation of the matrix is never considered, which can give benefits in terms of quantum gate complexity. In this work, we exploit the special structural property of a dense Vandermonde matrix by decomposing it into several sparse matrices and then use them for a complexity-efficient eigenvalue spectrum estimation in a quantum framework. In this context, we propose an iterative quantum simulation framework for a Vandermonde-structured Hamiltonian called quantum Vandermonde simulation (QVS). The proposed approach requires fewer quantum gate operations compared to the standard QHS of a dense Vandermonde matrix. In this work, we have considered a large antenna array system, that exploits a delay Vandermonde matrix (DVM) to create a multi-beam beamformer using the proposed low complex quantum architecture. Extensive theoretical and numerical analysis in terms of complexity and other quality parameters have been conducted to show the benefit of the proposed method.

A Quantum Algorithm for Solving Eigenproblem of the Laplacian Matrix of a Fully Connected Weighted Graph

AbstractSolving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix operations. Here, an efficient quantum algorithm is proposed to solve it. Specifically, the optimal Hamiltonian simulation technique based on the block‐encoding framework is adopted to implement the quantum simulation of the Laplacian matrix. Then, the eigenvalues and eigenvectors of the Laplacian matrix are extracted by the quantum phase estimation algorithm. The core of this entire algorithm is to construct a block‐encoding of the Laplacian matrix. To achieve this, how to construct block‐encoding of operators containing the information of the weight matrix and the degree matrix, respectively are shown in detail, and the block‐encoding of the Laplacian matrix is further obtained. Compared with its classical counterpart, this algorithm has a polynomial speedup on the number of vertices and an exponential speedup on the dimension of each vertex. It is also shown that this algorithm can be extended to solve the eigenproblem of symmetric (non‐symmetric) normalized Laplacian matrix.

Lax dynamics for Cartan decomposition with applications to Hamiltonian simulation

Abstract Simulating the time evolution of a Hamiltonian system on a classical computer is hard—The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge—Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution, is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. Within the integration errors that is controllable, this approach gives rise to a genuine unitary synthesis that not only is numerically feasible, but also can be utilized to gauge the quality of results produced by other means, and extend the knowledge to a wide range of applications. This paper aims at establishing the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics.