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arXiv:0801.3800 · Physical Review A

Nonperturbative k -body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins

Jacob Biamonte

Published May 23, 2008• 120 citations
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Abstract

An algebraic method has been developed which allows one to engineer several energy levels including the low-energy subspace of interacting spin systems. By introducing ancillary qubits, this approach allows $\mathit{k}$-body interactions to be captured exactly using two-body Hamiltonians. Our method works when all terms in the Hamiltonian share the same basis and has no dependence on perturbation theory or the associated large spectral gap. Our methods allow problem instance solutions to be embedded into the ground energy state of Ising spin systems. Adiabatic evolution might then be used to place a computational system into its ground state.

DOI
10.1103/physreva.77.052331

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Publisher's Note: Nonperturbative k -body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins [Phys. Rev. A77, 052331 (2008)]

Jacob Biamonte

Physical Review A•Jun 5, 2008•2 citations

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arXiv:1802.02285•Feb 7, 2018•3 citations
An algebraic method has been developed which allows one to engineer several energy levels including the low-energy subspace of interacting spin systems. By introducing ancillary qubits, this approach allows k\mathit{k}k-body interactions to be captured exactly using two-body Hamiltonians. Our method works when all terms in the Hamiltonian share the same basis and has no dependence on perturbation theory or the associated large spectral gap. Our methods allow problem instance solutions to be embedded into the ground energy state of Ising spin systems. Adiabatic evolution might then be used to place a computational system into its ground state.
An algebraic method has been developed which allows one to engineer several energy levels including the low-energy subspace of interacting spin systems. By introducing ancillary qubits, this approach allows k-body interactions to be captured exactly using 2-body Hamiltonians. Our method works when all terms in the Hamiltonian share the same basis and has no dependence on perturbation theory or the associated large spectral gap. Our methods allow problem instance solutions to be embedded into the ground energy state of Ising spin systems. Adiabatic evolution might then be used to place a computational system into it's ground state.
We present an intuitive compositional theory from which one is able to predict and also to control the ground state manifold (and higher energy excitations) of interacting spin systems governed by variants of tunable Ising models, hence giving precise control over the apriori additive structure of Hamiltonian composition. This compositional theory is given in terms of string diagrams: these results were made possible by mapping a variant of the Boolean F2-calculus onto spins and synthesizing modern ideas appearing in Category Theory, Coalgebras, Classical Network Theory and Graphical Calculus. Specifically, we present an algebraic method which allows one to explicitly engineer several energy levels including the low-energy subspace of interacting spin systems. We call this new framework: Ground State Spin Calculus, and in the first instance, the theory requires interactions of up to third order (3body). By introducing ancillary qubits, we present a novel approach allowing k-body interactions to be captured exactly using only two-body Hamiltonians [Biamonte, Phys. Rev. A 77(5), 052331 (2008)]. Our reduction method has no dependence on perturbation theory or the associated large spectral gap and allows for problem instance solutions to be embedded into the ground energy state of Ising spin systems. This could have important applications for future technology as adiabatic quantum evolution might be used to place such a computational system into it’s ground state.
The need for Hamiltonians with many-body interactions arises in various applications of quantum computing. However, interactions beyond two-body are difficult to realize experimentally. Perturbative gadgets were introduced to obtain arbitrary many-body effective interactions using Hamiltonians with at most two-body interactions. Although valid for arbitrary kkk-body interactions, their use is limited to small kkk because the strength of interaction is kthk\mathrm{th}kth order in perturbation theory. In this paper we develop a nonperturbative technique for obtaining effective kkk-body interactions using Hamiltonians consisting of at most lll-body interactions with l<kl<kl<k. The idea is to first apply a unitary transformation on the system plus ancilla qubits (possibly using a gate-based device), then evolve with a new Hamiltonian which is more local than the original one (using an analog device), and finally reverse the unitary transformation. The net effect of this procedure is shown to be equivalent to evolving the system with the original nonlocal Hamiltonian. This technique does not suffer from the aforementioned shortcoming of perturbative methods and requires only one ancilla qubit for each kkk-body interaction irrespective of the value of kkk. It works best for Hamiltonians with a few many-body interactions involving a large number of qubits and can be used together with perturbative gadgets to embed Hamiltonians of considerable complexity in proper subspaces of two-local Hamiltonians. We describe how our technique can be implemented in a hybrid (gate-based and adiabatic) as well as solely adiabatic quantum computing scheme.
Hamiltonian simulations are key subroutines in adiabatic quantum computation and quantum many-body physics, where quantum dynamics often happen in the low-energy sector. Previous studies have shown that the low-energy assumption can reduce the resource requirements of standard time-independent Hamiltonian simulation algorithms. However, whether such advantages extend to time-dependent Hamiltonian simulation remains open. In this paper, we consider the adiabatic regime where the relevant low-energy subspace is spanned by a fixed number of low-energy eigenstates and separated from the rest of the spectrum by a gap. We show that, for simulating spin Hamiltonians by product formulas, the explicit system size dependence in the leading commutator-scaling term can be replaced by a low-energy scale up to logarithmic factors. Technically, we derive the low-energy simulation error with commutator scaling for product formulas by leveraging adiabatic perturbation theory to analyze the time-variant energy spectrum of the underlying Hamiltonian. We further conduct numerical experiments on adiabatic state preparation of an illustrative example system to support our theoretical findings. Finally, we prove a lower bound of query complexity for generic time-dependent Hamiltonian simulations.
A central challenge in the successful implementation of adiabatic quantum algorithms is to maintain the quantum adiabaticity during the entire evolution. However, the energy gap between the ground and the excited states of interacting many-body systems often decreases quickly with the number of qubits, and the quantum adiabaticity can be severely impaired. Despite numerous previous efforts, a practical method to preserve the quantum adiabaticity has yet to be developed. Here we present a universal approach to enhancing the quantum adiabaticity via cavity or circuit QED. By coupling an adiabatic quantum computer to an ancilla cavity, the coupled system can exhibit a bistable regime with bifurcation points, where the time evolution becomes very slow. Utilizing these generic nonlinear features, we show that the energy gap of the adiabatic quantum computer can be positioned between the bifurcation points, which results in strongly-enhanced quantum adiabaticity. We then apply this method to a quantum two-level system, an Exact Cover problem, and a transverse field Ising model. In contrast to previous works, this approach does not require the spectral knowledge of the quantum system or the construction of unphysical interactions and can be applied to a vast variety of adiabatic quantum processes.