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International Journal of Quantum Information

GENERAL CIRCUITS FOR INDIRECTING AND DISTRIBUTING MEASUREMENT IN QUANTUM COMPUTATION

M. GUPTA, Anirban Pathak, R. Srikanth, K. PANIGRAHI

Published Aug 1, 2007• 22 citations
Publisher

Abstract

A question of theoretical and practical interest is how a quantum system may be measured indirectly by means of an ancilla that interacts with it, and furthermore, how a system of ancillas may be used to implement a coherent measurement of spatially separated qudits. We provide general circuits that can be used to implement such measurements. These circuits are relevant to quantum error correction, measurement-based quantum computation and Bell state discrimination across a quantum network involving multiple parties. The last mentioned problem is treated in detail. Our circuitry can also help to optimize the quantum communication complexity for performing measurements in distributed quantum computing.

DOI
10.1142/s0219749907003092

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A question of theoretical and practical interest is how a quantum system may be measured indirectly by means of an ancilla that interacts with it, and furthermore, how a system of ancillas may be used to implement a coherent measurement of spatially separated qudits. We provide general circuits that can be used to implement such measurements. These circuits are relevant to quantum error correction, measurement-based quantum computation and Bell state discrimination across a quantum network involving multiple parties. The last mentioned problem is treated in detail. Our circuitry can also help to optimize the quantum communication complexity for performing measurements in distributed quantum computing.
An ancilla-based single particle measurement scheme is illustrated for Bell state discrimination which does not lead to the collapse of the wave function.It enables the extraction of partial information, e.g., parity and phase independently, that generalizes to multi-particle entangled states of qubits, which also applied to qudits.It can be used for error correction in a quantum circuit architecture involving basic gates like Hadamard and Controlled-NOT.It helps unravel the syndrome operators and, what is more, in the multi-partite state that the individual constituents lack.
Many quantum algorithms require ancilla qubits and deep circuits, yet adding and interacting with them are difficult and resource-intensive. Moreover, the high-dimensional properties of quantum systems have been demonstrated to simplify quantum circuits while enhancing efficiency. We propose a novel “Dimension Expansion and Partial Measurement” scheme for quantum information processing based on qudits’ multi-level nature. Given a d-dimensional qudit (expandable to d × r dimensions) and local single-qudit operations in d and r dimensions, we can introduce r-dimensional equivalent ancilla qudits (logically introduced via dimension expansion) and enable quantum operations in deep circuits across d and r dimensions. With successfully generated photons, our scheme efficiently cascades two-qubit/qudit gates with unity theoretical success probability; existing schemes are either probabilistic or need additional particles. In particular, this scheme can be extended to multiple ancilla qudits with measurement feedforward by the “Partially Measure-Reexpand” procedure. This leads to a resource-efficient universal high-dimensional measurement-based quantum computation scheme. On a silicon-photonic chip, starting from 2 qubits (each expandable to a four-dimensional qudit), we implemented a measurement-based quantum computation system equivalent to 5 qubits and experimentally validated our scheme. The proposed scheme applies to simplifying and scaling up various quantum systems.
Although qubits are the leading candidate for the basic elements in a quantum computer, there are also a range of reasons to consider using higher-dimensional qudits or quantum continuous variables (QCVs). In this paper, we use a general ``quantum variable'' formalism to propose a method of quantum computation in which ancillas are used to mediate gates on a well-isolated ``quantum memory'' register and which may be applied to the setting of qubits, qudits (for d>2d>2d>2), or QCVs. More specifically, we present a model in which universal quantum computation may be implemented on a register using only repeated applications of a single fixed two-body ancilla-register interaction gate, ancillas prepared in a single state, and local measurements of these ancillas. In order to maintain determinism in the computation, adaptive measurements via a classical feed forward of measurement outcomes are used, with the method similar to that in measurement-based quantum computation (MBQC). We show that our model has the same hybrid quantum-classical processing advantages as MBQC, including the power to implement any Clifford circuit in essentially one layer of quantum computation. In some physical settings, high-quality measurements of the ancillas may be highly challenging or not possible, and hence we also present a globally unitary model which replaces the need for measurements of the ancillas with the requirement for ancillas to be prepared in states from a fixed orthonormal basis. Finally, we discuss settings in which these models may be of practical interest.
We show that coherent error correction with redundant syndrome extraction is feasible even when realistic error models for the ancilla qubits are used. This is done by means of a fault-tolerant circuit that has a favorable error threshold for a conservative error model. It is optimal when Toffoli gates are available, as they are in many implementations. The possibility of coherent error correction is important for those quantum computing platforms in which measurements in the course of the computation are undesirable.
Low depth measurement-based quantum computation with qudits (ddd-level systems) is investigated and a precise relationship between this powerful model and qudit quantum circuits is derived in terms of computational depth and size complexity. To facilitate this investigation a qudit `unbounded fan-out' circuit model, in which a qudit may be quantum-copied into an arbitrary number of ancillas in a single time-step, is introduced and shown to be capable of implementing interesting nnn-qudit unitaries in constant depth. A procedure for reducing the quantum computational depth in the measurement-based model is then proposed and using this it is then shown that there is a logarithmic depth separation between the depth complexity of qudit measurement-based computation and circuits composed of gates act on a bounded number of qudits. The relationship is made precise by showing that the depth complexity of the qudit measurement-based model is exactly equivalent to that of unbounded fan-out circuits. These results illustrate that the well-known advantages inherent in qubit measurement-based quantum computation are also applicable to the higher-dimensional generalisation. As qudits are both naturally available and have been shown to provide fundamental advantages over binary logic encodings, this then suggests that the qudit measurement-based model is a particularly appealing paradigm for universal quantum computation.