Quantum encodings in spin systems and harmonic oscillators

Abstract

We show that higher-dimensional versions of qubits, or qudits, can be encoded into spin systems and into harmonic oscillators, yielding important advantages for quantum computation. Whereas qubit-based quantum computation is adequate for analyses of quantum vs classical computation, in practice qubits are often realized in higher-dimensional systems by truncating all but two levels, thereby reducing the size of the precious Hilbert space. We develop natural qudit gates for universal quantum computation, and exploit the entire accessible Hilbert space. Mathematically, we give representations of the generalized Pauli group for qudits in coupled spin systems and harmonic oscillators, and include analyses of the qubit and the infinite-dimensional limits.PACS numbers: 03.67. Lx, Quantum computation may be able to perform certain tasks more efficiently than a classical computer; for example, Shor's algorithm [1] for factoring prime numbers on a quantum computer is exponentially faster than any known algorithm on a classical computer. The standard model of a quantum computer involves coupling together two-level quantum systems (qubits) such that the Hilbert space of the system grows exponentially in the number of qubits.A major obstacle to universal quantum computing is the limit on the number of coupled qubits that can be achieved in a physical system [2]. The use of ddimensional, or qudit, quantum computing enables a much more compact and efficient information encoding than for qubit computing. Qudit quantum information processing employs fewer coupled quantum systems: a considerable advantage for the experimental realization of quantum computing. The harmonic oscillator is a system that naturally provides qudits as quanta in its energy spectrum. Qubits are obtained by restricting the dynamics to just two of these quanta, namely the vacuum state |0 and the first excited state |1 ; e.g., photons in cavity QED [3] and interferometry [4]. However, the control of entanglement in larger Hilbert spaces is now feasible (e.g., orbital angular momentum states of photons [5]). Our aim is to show that the restriction to two-dimensional Hilbert spaces is not necessary and that higher-dimensional Hilbert spaces are an advantage, particularly when the number of achievable coupled systems is limited and entanglement between systems with larger Hilbert spaces is physically possible.A quantum computer also requires gates, realized as the unitary evolution under some Hamiltonian. For qubits, a universal set of gates is given by arbitrary SU(2) rotations of a single qubit along with some nonlinear coupling transformation between adjacent qubits generated by a two-qubit Hamiltonian [6]. For qudit quantum computation, the issue of creating a universal set of gates is more involved. In particular, it is not possible to treat coupled qudits as a collection of qubits, because (typically) one does not have access to "pairwise" Hamiltonians between two arbitrary levels of coupled qudits. For example, in a system of coupled oscillators realize...

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