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arXiv:1011.2984 · New Journal of Physics

Ancilla-based quantum simulation

Katherine L. Brown, Suvabrata De, Viv M. Kendon, William J. Munro

Published Sep 14, 2011• 19 citations
arXivHTMLPDF

Abstract

We consider the simulation of the Bardeen, Cooper and Schrieffer (BCS) Hamiltonian, a model of low-temperature superconductivity, on a quantum computer. In particular, we consider conducting the simulation on the qubus quantum computer, which uses a continuous variable ancilla to generate interactions between qubits. We demonstrate an O ( N 3 ) improvement over previous studies conducted on an NMR computer (Wu et al 2002 Phys. Rev. Lett. 89 057904 and Brown et al 2006 Phys. Rev. Lett. 97 050504) for the nearest-neighbour and completely general cases. We then proceed to show methods for minimizing the number of operations needed per time step using the qubus in three cases: the completely general case, the case of exponentially decaying interactions and the case of fixed range interactions. We make these results controlled on an ancilla qubit so that we can apply the phase estimation algorithm, and hence show that when N ⩾5, our qubus simulation requires significantly fewer operations than a similar simulation conducted on an NMR computer.

DOI
10.1088/1367-2630/13/9/095007

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We consider the simulation of the Bardeen, Cooper and Schrieffer (BCS) Hamiltonian, a model of low-temperature superconductivity, on a quantum computer. In particular, we consider conducting the simulation on the qubus quantum computer, which uses a continuous variable ancilla to generate interactions between qubits. We demonstrate an O ( N 3 ) improvement over previous studies conducted on an NMR computer (Wu et al 2002 Phys. Rev. Lett. 89 057904 and Brown et al 2006 Phys. Rev. Lett. 97 050504) for the nearest-neighbour and completely general cases. We then proceed to show methods for minimizing the number of operations needed per time step using the qubus in three cases: the completely general case, the case of exponentially decaying interactions and the case of fixed range interactions. We make these results controlled on an ancilla qubit so that we can apply the phase estimation algorithm, and hence show that when N ⩾5, our qubus simulation requires significantly fewer operations than a similar simulation conducted on an NMR computer.
We describe in detail how a qubus quantum computer using a continuous variable ancilla to control the qubits can provide significant efficiency savings for quantum simulation. We illustrate this by using as an example the simulation of the Bardeen, Cooper and Schrieffer (BCS) Hamiltonian, a model of low-temperature superconductivity. We demonstrate an O(Ns) improvement over previous studies conducted on an NMR computer [1, 2] for nearest- neighbour and completely general cases. We make these results controlled on an ancilla qubit so that we can apply the phase estimation algorithm, and hence show that when N ≥ 5, our qubus simulation requires significantly fewer operations than a similar simulation conducted on an NMR computer.
In this thesis I explore using the qubus for quantum computing. The qubus is an architecture of quantum computing, where a continuous variable ancilla is used to generate operations between matter qubits. I concentrate on using the qubus for two purposes - quantum simulation, and generating cluster states. Quantum simulation is the idea of using a quantum computer to simulate a quantum system. I focus on conducting a simulation of the BCS Hamiltonian. I demonstrate how to perform the necessary two qubit operations in a controlled fashion using the qubus. In particular I demonstrate an O(N3) saving over an implementation on an NMR computer, and a factor of 2 saving over a naıve technique. I also discuss how to perform the quantum Fourier transform on the qubus quantum computer. I show that it is possible to perform the quantum Fourier transform using just, 24⌊N/2⌋ + 7N − 6, this is an O(N) saving over a naıve method. In the second part of the thesis, I move on, and consider generating cluster states using the qubus. A cluster state, is a universal resource for one-way or measurement-based computation. In one-way computation, the pre-generated, entangled resource is used to perform calculations, which only require local corrections and measurement. I demonstrate that the qubus can generate cluster states deterministically, and in a relatively short time. I discuss several techniques of cluster state generation, one of which is optimal, given the physical architecture we are using. This can generate an n × m cluster in only 3nm − 2n − 2m + 4 operations. The alternative techniques look at generating a cluster using layers or columns, allowing it to be built dynamically, while the cluster is used to perform calculations. I then move on, and discuss problems with error accumulation in the generation process.
Cluster states can be used to perform measurement-based quantum computation. The cluster state is a useful resource, because once it has been generated only local operations and measurements are needed to perform universal quantum computation. In this paper, we explore techniques for quickly and deterministically building a cluster state. In particular, we consider generating cluster states on a qubus quantum computer, a computational architecture which uses a continuous variable ancilla to generate interactions between qubits. We explore several techniques for building the cluster, with the number of operations required depending on whether we allow the ability to destroy previously created controlled-phase links between qubits. In the case where we cannot destroy these links, we show how to create an $n\ifmmode\times\else\texttimes\fi{}m$ cluster using just $3nm\ensuremath{-}2n\ensuremath{-}\ensuremath{\lceil}3m/2\ensuremath{\rceil}+3$ operations. This gives more than a factor of 2 savings over a naive method. Further savings can be obtained if we include the ability to destroy links, in which case we only need $\ensuremath{\lceil}\frac{1}{3}(8nm\ensuremath{-}4n\ensuremath{-}4m\ensuremath{-}8)\ensuremath{\rceil}$ operations. Unfortunately, the latter scheme is more complicated so choosing the correct order to interact the qubits is considerably more difficult. A halfway scheme, that keeps a modular generation but saves additional operations over never destroying links requires only $3nm\ensuremath{-}2n\ensuremath{-}2m+4$ operations. The first scheme and the last scheme are the most practical for building a cluster state because they split up the generation into the repetition of simple sections.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.