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arXiv:quant-ph/0109063 · Quantum Information and Computation

Universal simulation of Hamiltonians using a finite set of control operations

Paweł Wocjan, Martin Rötteler, Dominik Janzing, T. Beth

Published Feb 1, 2002• 45 citations
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Abstract

Any quantum system with a non-trivial Hamiltonian is able to simulate any other Hamiltonian evolution provided that a sufficiently large group of unitary control operations is available. We show that there exist finite groups with this property and present a sufficient condition in terms of group characters. We give examples of such groups in dimension 2 and 3. Furthermore, we show that it is possible to simulate an arbitrary bipartite interaction by a given one using such groups acting locally on the subsystems.

DOI
10.26421/qic2.2-3

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Any quantum system with a non-trivial Hamiltonian is able to simulate any other Hamiltonian evolution provided that a sufficiently large group of unitary control operations is available. We show that there exist finite groups with this property and present a sufficient condition in terms of group characters. We give examples of such groups in dimension 2 and 3. Furthermore, we show that it is possible to simulate an arbitrary bipartite interaction by a given one using such groups acting locally on the subsystems.
We describe a broad dynamical-algebraic framework for analyzing the quantum control properties of a set of naturally available interactions. General conditions under which universal control is achieved over a set of subspaces/subsystems are found in terms of the representation theory of the group UA{\mathcal{U}}_{A}UA​ of allowed quantum evolutions. In some cases universal control can be achieved in all the state-space sectors carrying an irreducible action of UA.{\mathcal{U}}_{A}.UA​. All known physical examples of universal control on subspaces/systems are related to the framework developed here. Implications for quantum-information processing are discussed.
When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that are not universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator expansion contains only terms coupling an odd number of systems, as identified by Bremner et al. [Phys. Rev. A 69, 012313 (2004)]. We show that in all other cases entangling many-qudit Hamiltonians are universal for simulation.
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.
Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. For the case where this set consists of only products of Pauli operators, known as Pauli strings, we provide a necessary and sufficient condition for it to generate su(2n)\mathfrak{su}(2^n)su(2n), i.e., to be universal for quantum computation on nnn qubits. When combining Pauli strings with a general Hamiltonian, we show a sufficient (and in certain circumstances even necessary) condition for universality based on the Pauli-basis expansion of the Hamiltonian. As an application of these results, we prove two corollaries: (i) a necessary and sufficient condition for the universality of a general Hamiltonian given arbitrary single-qubit control on all qubits, and (ii) the universality of an XYZ Heisenberg Hamiltonian with local control of just two adjacent qubits.
What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? Dodd et al. [Phys. Rev. A 65, 040301(R) (2002)] provided a partial solution to this problem in the form of an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling N-qubit Hamiltonian, and local unitaries. We extend this result to the case where the component systems are qudits, that is, have D dimensions. As a consequence we explain how universal quantum computation can be performed with any fixed two-body entangling N-qudit Hamiltonian, and local unitaries.