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arXiv:quant-ph/0107035 · Physical Review A

Optimal simulation of two-qubit Hamiltonians using general local operations

C. H. Bennett, J. I. Cirac, Matthew Leifer, Debbie Leung, Noah Linden, Sandu Popescu, Guifré Vidal

Published Jul 19, 2002• 110 citations
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Abstract

We consider the simulation of the dynamics of one nonlocal Hamiltonian by another, allowing arbitrary local resources but no entanglement or classical communication. We characterize notions of simulation, and proceed to focus on deterministic simulation involving one copy of the system. More specifically, two otherwise isolated systems A and B interact by a nonlocal Hamiltonian $H\ensuremath{\ne}{H}_{A}{+H}_{B}.$ We consider the achievable space of Hamiltonians ${H}^{\ensuremath{'}}$ such that the evolution ${e}^{\ensuremath{-}{\mathrm{iH}}^{\ensuremath{'}}t}$ can be simulated by the interaction H interspersed with local operations. For any dimensions of A and B, and any nonlocal Hamiltonians H and ${H}^{\ensuremath{'}},$ there exists a scale factor s such that for all times t the evolution ${e}^{\ensuremath{-}{\mathrm{iH}}^{\ensuremath{'}}\mathrm{st}}$ can be simulated by H acting for time t interspersed with local operations. For two-qubit Hamiltonians H and ${H}^{\ensuremath{'}},$ we calculate the optimal s and give protocols achieving it. The optimal protocols do not require local ancillas, and can be understood geometrically in terms of a polyhedron defined by a partial order on the set of two-qubit Hamiltonians.

DOI
10.1103/physreva.66.012305

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We consider the simulation of the dynamics of one nonlocal Hamiltonian by another, allowing arbitrary local resources but no entanglement or classical communication. We characterize notions of simulation, and proceed to focus on deterministic simulation involving one copy of the system. More specifically, two otherwise isolated systems A and B interact by a nonlocal Hamiltonian Hnot equalH(A)+H-B. We consider the achievable space of Hamiltonians H-' such that the evolution e(-iH')t can be simulated by the interaction H interspersed with local operations. For any dimensions of A and B, and any nonlocal Hamiltonians H and H-', there exists a scale factor s such that for all times t the evolution e(-iH')st can be simulated by H acting for time t interspersed with local operations. For two-qubit Hamiltonians H and H-', we calculate the optimal s and give protocols achieving it. The optimal protocols do not require local ancillas, and can be understood geometrically in terms of a polyhedron defined by a partial order on the set of two-qubit Hamiltonians.
Consider a set of N systems and an arbitrary interaction Hamiltonian H that couples them. We investigate the use of local operations and classical communication (LOCC), together with the Hamiltonian H, to simulate a unitary evolution of the N systems according to some other Hamiltonian ${H}^{\ensuremath{'}}.$ First, we show that the most general simulation using H and LOCC can also be achieved, with the same time efficiency, by just interspersing the evolution of H with local unitary manipulations (LU) of each system and a corresponding local ancilla (in a so-called LU+anc. protocol). Thus, the ability to make local measurements and to communicate classical information does not help in nonlocal Hamiltonian simulation. Second, we show that both for the case of two d-level systems (d>2),(d>2),(d>2), or for that of a setting with more than two systems (N>2),(N>2),(N>2), LU+anc. protocols are more powerful than LU protocols. Therefore local ancillas are a useful resource for nonlocal Hamiltonian simulation. Third, we use results of majorization theory to explicitly solve the problem of optimal simulation of two-qubit Hamiltonians using LU (equivalently, LU+anc., LO, or LOCC).
Consider two quantum systems A and B interacting according to a product Hamiltonian H=H/sub A//spl ominus/H/sub B/. We show that any two such Hamiltonians can be used to simulate each other reversibly (i.e., without efficiency losses) with the help of local unitary operations and local ancillas. Accordingly, all nonlocal features of a product Hamiltonian - including the rate at which it can be used to produce entanglement, transmit classical or quantum information, or simulate other Hamiltonians - depend only upon a single parameter. We identify this parameter and use it to obtain an explicit expression for the entanglement capacity of all product Hamiltonians. Finally, we show how the notion of simulation leads to a natural formulation of measures of the strength of a nonlocal Hamiltonian.
We study when a nonlocal unitary operation acting on two d-level systems can probabilistically simulate another one when arbitrary local operations and classical communication are allowed. We provide necessary and sufficient conditions for the simulation to be possible. Probabilistic interconvertability is used to define an equivalence relation between gates. We show that this relation induces a finite number of classes that we identify. In the case of two qubits, two classes of nonlocal operations exist. We show how the representatives of these classes, CNOT and SWAP, can be deterministically converted into any operation of its class and calculate the optimal probability of the reverse process.
Digital-analog is a quantum computational paradigm that employs the natural interaction Hamiltonian of a system as the entangling resource, combined with single qubit gates, to implement universal quantum operations. As in the case of its digital gate-based counterpart, designing digital-analog circuits that employ optimal quantum resources often requires an exceedingly large classical computational time. In this work we find a suboptimal solution to this exponentially large problem, showing that it can be solved within polynomial computational time. In particular, we provide an exact solution for the problem of expressing arbitrary two-body Hamiltonians as the sum of local unitary transformations of an arbitrary Ising Hamiltonian, with the total number of required terms being at most quadratic in system size. This allows us to design a digital-analog simulation protocol that avoids employing numerical optimization over a large parameter space at the preprocessing stage, minimizing computational resources and allowing for further scaling.
What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? It has been shown that all two-body Hamiltonian evolutions can be simulated using any fixed two-body entangling n-qubit Hamiltonian and fast local unitaries. By entangling we mean that every qubit is coupled to every other qubit, if not directly, then indirectly via intermediate qubits. We extend this study to the case where interactions may involve more than two qubits at a time. We find necessary and sufficient conditions for an arbitrary n-qubit Hamiltonian to be dynamically universal, that is, able to simulate any other Hamiltonian acting on n qubits, possibly in an inefficient manner. We prove that an entangling Hamiltonian is dynamically universal if and only if it contains at least one coupling term involving an even number of interacting qubits. For odd entangling Hamiltonians, i.e., Hamiltonians with couplings that involve only an odd number of qubits, we prove that dynamic universality is possible on an encoded set of $n\ensuremath{-}1$ logical qubits. We further prove that an odd entangling Hamiltonian can simulate any other odd Hamiltonian and classify the algebras that such Hamiltonians generate. Thus, our results show that up to local unitary operations, there are only two fundamentally different types of entangling Hamiltonian on n qubits. We also demonstrate that, provided the number of qubits directly coupled by the Hamiltonian is bounded above by a constant, our techniques can be made efficient.