Session A09: Algorithms and Architecture for Quantum Information I

Finding Navier-Stokes fluid flows through quantum computung

We present a quantum algorithm that solves an arbitrary set of coupled non-linear partial differential equations and show how it can be used to solve the governing equations for a Navier-Stokes fluid. To test the algorithm we examine the problem of inviscid, compressible flow through a convergent-divergent nozzle. We numerically simulate application of the algorithm to find the steady-state flow when a shockwave is and is not present in the divergent part of the nozzle. In each case excellent agreement is found between the output of the quantum simulation and the exact analytical solution, with the simulation successfully capturing the shockwave when present. We compare the computational cost of the quantum algorithm to that of deterministic and random classical algorithms; discuss future applications as well as the potential long-term significance of quantum computing for the fluid dynamics community.   

Quantum eigenvalue estimation via time series analysis

We present an efficient method for estimating the eigenvalues of a Hamiltonian H from the expectation values of the evolution operator for various times. For a given quantum state ρ, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of ρ in those eigenstates of H associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter. Unlike the well-known quantum phase estimation algorithm that uses the quantum Fourier transform, our method does not require large ancillary systems, large sequences of controlled operations, or preserving coherence between experiments, and is therefore more attractive for near-term applications. The output of our method can be used to compute spectral properties of H and other expectation values efficiently.   

A robust algorithm for finding phase factors in quantum signal processing

Quantum Signal Processing (QSP) provides a general way to implement matrix functions on quantum computers. The algorithm can be efficiently used to solve quantum linear systems, to perform Hamiltonian simulation, and to prepare Gibbs ensembles, among other applications. QSP can exactly encode a degree-d polynomial transformation of a matrix using d+1 phase factors. However, the current strategies for solving for the phase factors of a given function can be numerically unstable. We present an efficient method to find the phase factors for a general real function, and demonstrate the performance for solving linear systems and eigenvalue problems.   

Finding symmetry-broken ground states with variational quantum algorithms

One of the most promising applications for near-term intermediate scale quantum computers (NISQ) is the preparation of the true ground state of strongly correlated electron systems. Besides the ground-state energy the properties of interest of the ground state are its broken symmetries and the corresponding phases of the system.

We study the preparation of broken symmetry ground states on a gate based quantum computer with different variational algorithms, including the variational Hamiltonian ansatz (VHA) with initial state preparation and extensions to the standard VHA.
The two-dimensional Hubbard model is used as a toy model, to compare the variational algorithms to each other and to exact diagonalisation.
To this end, we simulate the full algorithm including initialization and read-out running on a gate-based quantum computer. We use a hardware model based on the gates available in currently available quantum computers.   

Strategies for digital quantum simulation of bosons

Many prominent bosonic simulation problems are thought to be intractable on a classical computer, including the Bose-Hubbard model, quantum photonics, and molecular vibronics. The behavior of such systems would be efficiently studied on a quantum computer. Before such a simulation is performed, one must choose how to encode the bosonic degrees of freedom into a set of qubits. We present a general methodology for encoding truncated bosons into arrays of qubits, and consider several encoding types. We study the quantum operations and qubit counts for local and composite operators. Importantly, we also consider the utility of interconverting between mappings in the middle of a simulation. These methods lower the quantum resource requirements compared to previous encoding strategies, which may allow for larger problems to be simulated on near-term quantum devices.   

A Method of Determining Excited-States for Quantum Computation

The calculation of molecular ground-state and excited-state energies, or more generally the energy spectra of chemical and material systems, is an application of great interest in a gate-based quantum computational model. In this contribution, we propose a phenomenological approach to the calculation of low-lying excited-states of a given problem Hamiltonian. Specifically, a method is presented in which the ground-state subspace is projected out of a Hamiltonian representation. As a result of this projection, an effective Hamiltonian is constructed where its ground-state coincides with an excited-state of the original problem. Thus, low-lying excited-state energies can be calculated using existing hybrid quantum-classical techniques and variational algorithm(s) for determining ground-state. The method is shown to be fully valid for the H2 molecule. In addition, conditions for the method’s success are discussed in terms of classes of Hamiltonians. A discussion on the broad impact of this method in the era of NISQ devices will also be presented.   

Quantum simulation by qubitization without Toffoli gates

Qubitization is a modern approach to estimate Hamiltonian eigenvalues without simulating its time evolution. While in this way approximation errors are avoided, its resource and gate requirements are more extensive: qubitization requires additional qubits to store information about the Hamiltonian, and Toffoli gates to probe them throughout the routine. Recently, it was shown that storing the Hamitlonian in a unary representation can alleviate the need for such gates in one of the qubitization subroutines. Building on that principle, we develop an entirely new decomposition of the entire algorithm: without Toffoli gates, we can encode the Hamiltonian into qubits within logarithmic depth.   

Emulation of fractional quantum Hall states with existing quantum hardware

Emulating strongly correlated phases of matter with existing or near-term quantum hardware is of considerable interest. Here we present quantum circuits to generate both the ground and quasiparticle states corresponding to the Laughlin's $\nu=1/3$ fractional quantum Hall state in the second-quantized representation. Our circuit depth is linear in the number of qubits. We identify experimentally accessible correlation functions as signatures of this state and discuss the quantum hardware implementation.   

Computing partition functions in the one clean qubit model

In this talk we will present a method to approximate partition functions of quantum systems using mixed-state quantum computation. For non-negative Hamiltonians, our method runs on average in time almost linear in (M/(ε_relZ))², where M is the dimension of the quantum system, Z is the partition function, and ε_rel is the relative precision. It is based on approximations of the exponential operator as linear combinations of certain operations related to block-encoding of Hamiltonians or Hamiltonian evolutions. The trace of each operation is estimated using a standard algorithm in the one clean qubit model. For large values of Z, our method may run faster than exact classical methods, whose complexities are polynomial in M. Using this method we will demonstrate that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that our method provides an exponential speedup.   

Reaction dynamics on a quantum computer

The study of quantum chemistry is expected to be a principal use of emergent quantum computing devices. The ability of quantum computers to efficiently provide highly accurate electronic structure data could have major consequences where such accuracy is required, such as in the prediction of the kinetics and dynamics of chemical reactions. However, such simulations often require vast numbers of electronic structure calculations, potentially exacerbating resource limitations of small-scale quantum devices.

In this talk, we present theoretical results characterising the quantum resources required for the simulation of reactions involving small numbers of light atoms through semi-classical trajectory simulation. We consider how this process can be aided by recent optimisations to variational quantum algorithms. Finally, we report simulation results and discuss experimental progress to this end.

  

A size-extensive scheme for variational quantum ansatzes without Trotter approximation

One of the most promising applications for quantum computers is simulating the low-energy states of complex quantum systems. In the near-term, this can be done with variational ansatzes. Such ansatzes should follow physical principles to ensure high performance, one of the key principles being size-extensivity. Unfortunately, digitizing such ansatzes into standard operations generally requires the use of the inexact Trotter expansion, which constrains the expected accuracy of the ansatz.

In this work, we resolve this conflict by developing a framework for physically motivated ansatzes, which is fundamentally digital and thereby involves no Trotter errors. Using the stabilizer formalism, we construct a family of digital ansatzes that provably cover the entire Hilbert space with a minimal number of parameters. We show how to compress such parent ansatzes into practical child ansatzes that target specific systems, following the principle of size-extensivity. For this purpose, we develop a convenient diagrammatic approach. We apply our method numerically to the quantum Ising chain, with good convergence outside the critical regime.   

Finding the ground state of the Hubbard model by variational methods on a quantum computer with gate errors

A key goal of digital quantum computing is the simulation of fermionic systems such as molecules or the Hubbard model. Unfortunately, for present and near-future quantum computers the use of quantum error correction schemes is still out of reach. Hence, the finite error rate limits the use of quantum computers to algorithms with a low number of gates. The variational Hamiltonian ansatz (VHA) has been shown to produce the ground state in good approximation in a manageable number of steps. Here we study explicitly the effect of gate errors on its performance. The VHA is inspired by the adiabatic quantum evolution under the influence of a time-dependent Hamiltonian, where the -- ideally short -- fixed Trotter time steps are replaced by variational parameters. The method profits substantially from quantum variational error suppression, e.g., unitary quasi-static errors are mitigated within the algorithm. We test the performance of the VHA when applied to the Hubbard model in the presence of unitary control errors on quantum computers with realistic gate fidelities.   

Quantum algorithm for spectral projection by measuring an ancilla iteratively

We propose a quantum algorithm for projecting a quantum system to eigenstates of any Hermitian operator, provided one can access the associated control-unitary evolution for the ancilla and the system. The procedure is iterative by preparing a fresh ancilla state, applying the controlled unitary, and then measuring the ancilla. There are some freedoms in the ancilla state parameter and the control unitary. We give examples to illustrate that the alogrithm works. Simulations of the procedure also show that the distribution of the projected eigenstates obeys the Born rule. This algorithm can be used as a subroutine in the quantum annealing procedure by measurement to drive the system to the ground state, and we also simulate this for a quantum spin chain.
  

Quantum Simulation of Quantum Z2 Gauge Theory demonstrated in a GPU Simulator

We outline a quantum simulation scheme of quantum Z2 gauge theory using quantum adiabatic algorithm implemented in terms of quantum circuit, and then demonstrate it in the classical simulator QuEST using a CUDA enabled GPU server. In particular, we obtained useful results in (3+1)-dimensional and (2+1)-dimensional theories. It is identified that the quantum phase transition is topological in both dimensions, and is first-order in (3+1) dimensions but second-order in (2+1) dimensions. High-performance classical simulation of quantum simulation, which may be dubbed pseudoquantum simulation, is not only a platform of developing quantum software, but also represents a new practical mode of computation.   

Term Grouping Techniques for VQE and Quantum Dynamics Circuits

Digital quantum simulations are among the most promising near-term applications of quantum computation. Variational Quantum Eigensolver and time evolution of quantum dynamics are two examples of such algorithms. However, the amount of required quantum resources typically do not scale favorably as the desired accuracy of the calculations increases. Both VQE and quantum dynamics circuits are represented by tensor products of Pauli matrices that are obtained from the second quantization form using transformation methods such as Jordan-Wigner or Bravyi-Kitaev. We demonstrate various grouping techniques that optimize the order of these tensor products, with the goal of optimizing the total quantum resource cost. For VQE circuits, we minimize the number of required measurement operations. For quantum dynamics circuits, we minimize the circuit depth and maximize its fidelity.