Quantum gates via continuous time quantum walks in multiqubit systems with non-local auxiliary states

Abstract

Non-local higher-energy auxiliary states have been successfully used to entangle pairs of qubits in different quantum computing systems. Typically a longer-span non-local state or sequential application of few-qubit entangling gates are needed to produce a non-trivial multiqubit gate. In many cases a single non-local state that span over the entire system is difficult to use due to spectral crowding or impossible to have. At the same time, many multiqubit systems can naturally develop a network of multiple nonlocal higher-energy states that span over few qubits each. We show that continuous time quantum walks can be used to address this problem by involving multiple such states to perform local and entangling operations concurrently on many qubits. This introduces an alternative approach to multiqubit gate compression based on available physical resources. We formulate general requirements for such walks and discuss configurations of non-local auxiliary states that can emerge in quantum computing architectures based on self-assembled quantum dots, defects in diamond, and superconducting qubits, as examples. Specifically, we discuss a scalable multiqubit quantum register constructed as a single chain with nearest-neighbor interactions. We illustrate how quantum walks can be configured to perform single-, two- and three-qubit gates, including Hadamard, Control-NOT, and Toffoli gates. Continuous time quantum walks on graphs involved in these gates are investigated.

Quantum walks as mathematical foundation for quantum gates

“…Appropriately designed continuous time quantum walks through such networks accumulate nontrivial phase faster then traditional entangling gates under the same condition but relying on only small part of such network each time. However these earlier derivations 29 were obtained in resonant and rotating wave approximations. The question of whether quantum walk description is valid in a more general case, when these approximations are inappropriate, was not resolved.…”

“…Both types of quantum walks were proven to provide similar quantum speedup as regular quantum computing. 30,41,42 Recently it was demonstrated 29 that continuous time quantum walks is a natural description of hardware quantum gates taking advantage of extended Hilbert space available in many qubit architectures targeting gate-based quantum computing. A collection of auxiliary states (states beyond the boolean computational domain), many of which participate in cross-qubit interactions, can be naturally viewed as non-boolean (not qubit-based) quantum network.…”

“…Note that each V ± ω still depend on time via slowly changing Φ(t), pulse envelop function, but this dependence is trivial, see Eq. (29).…”

“…Yet, design of quantum gates and quantum code compression typically aims to suppress this freedom by relying on deterministic sequences of controls micromanaging quantum evolution and trajectories. Recent analysis of few-qubit (entangling) quantum gates performed via continuous time quantum walks driven by classical field, 29 have shown that by allowing greater freedom for the quantum particle during multiqubit rotations one can significantly speedup entangling quantum gates. Yet, formulation of such gates have still been attached to chosen qubit architectures by relying on resonant approximation that is very specific to the actual physical system at hand.…”

“…Physical interactions necessary to carry and propagate entanglement enter via symmetry of edges (connections) independently of whether resonant approximation is used. The typical approximations, such as a rotating wave approximation [38][39][40] and a resonant approximation, 8,29 are naturally described withing the quantum walk approach and can be verified by specifically designed quantum walks. We show that when resonant approximation is appropriate, quantum coins are factored out and quantum gates are described by continuous time quantum walks.…”