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International Journal of Quantum Information

Measurement-based quantum computation cannot avoid byproducts

Tomoyuki Morimae

Published Jul 16, 2014• 7 citations
Publisher

Abstract

In the circuit model of quantum computation, a desired unitary gate can be implemented deterministically, whereas in the measurement-based model the unitary gate is implemented up to a byproduct operator. In order to compensate byproducts, following measurement angles must be adjusted, or classical results must be corrected. Such a feed-forwarding requires some classical processing and tuning of the measurement device, which cause the delay of computation and the additional decoherence. We show that if we respect the no-signaling principle, which is one of the most fundamental principles in physics, byproducts cannot be avoided in measurement-based quantum computation. Furthermore, we also show by using the idea of the quantum error correcting code that due to the no-signaling principle, not all byproducts are allowed in measurement-based quantum computation.

DOI
10.1142/s0219749914500269

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In the circuit model of quantum computation, a desired unitary gate can be implemented deterministically, whereas in the measurement-based model the unitary gate is implemented up to a byproduct operator. In order to compensate byproducts, following measurement angles must be adjusted, or classical results must be corrected. Such a feed-forwarding requires some classical processing and tuning of the measurement device, which cause the delay of computation and the additional decoherence. We show that if we respect the no-signaling principle, which is one of the most fundamental principles in physics, byproducts cannot be avoided in measurement-based quantum computation. Furthermore, we also show by using the idea of the quantum error correcting code that due to the no-signaling principle, not all byproducts are allowed in measurement-based quantum computation.
Measurement-based quantum computation is a novel model of quantum computing where universal quantum computation can be done with only local measurements on each particle of a quantum many-body state, which is called a resource state. One large difference of the measurement-based model from the circuit model is the existence of byproducts. In the circuit model, a desired unitary U can be implemented deterministically, whereas the measurement-based model implements BU, where B is an additional operator, which is called a byproduct. In order to compensate byproducts, following measurement angles must be adjusted. Such a feed-forwarding requires some classical processing and tuning of the measurement device, which cause the delay of computation and the additional decoherence. Is there any byproduct-free resource state? Here we show that if we respect the no-signaling principle, which is one of the most fundamental principles of physics, no universal resource state can avoid byproducts.
Measurements and feedforward enhance the power of shallow quantum circuits, enabling the deterministic implementation of global unitary operations and the preparation of long-range entangled states. We establish a general correspondence between all such protocols and quantum error-correcting codes: the circuit preceding the measurements acts as an encoder, and unitary feedforward can eliminate post-selection if and only if the measurement projectors are detectable errors on the codespace. This correspondence provides a common framework for state preparation and implementation of global unitaries with measurements and feedforward, turning both into a code-design problem. For stabilizer codes with Pauli measurements, although the resulting operation can be non-Clifford, we show that its nonstabilizerness originates entirely from the encoder. We overcome this restriction using non-Pauli measurements or non-additive codes, constructing protocols that generate long-range nonstabilizerness while requiring only single-qubit unitary corrections.
In Chapter 4 we introduced different quantum gates and quantum circuits and in Chapter 5 we described several quantum algorithms. So far the analysis of the algorithms and the circuits has been made with the assumption that the required unitary operations can be implemented exactly without any error. This is an abstraction. In an actual physical system that implements a quantum algorithm or a quantum circuit there are several sources of errors. The situation is similar in classical computation and to circumvent this problem error correcting codes are used in classical digital computing. Since qubits are more delicate and more susceptible to errors we need to introduce quantum error correcting codes. It is very important, as without such codes it will not be possible to build a scalable quantum computer. Such quantum error correcting codes are introduced in this chapter. However, we will not limit ourselves to the discussion of error correcting codes alone. We will also discuss more general aspects of errors and error correction, like decoherence, decoherence free subspace and fault-tolerance.
In order to stabilize quantum algorithms against decoherence one has to fulfill two requirements. Firstly, one has to develop an appropriate quantum error correcting code. Secondly, one has to find a set of suitable unitary quantum transformations acting on the physical qubits which preserve the properties of this error correcting quantum code and which allow the implementation of a universal set of quantum gates. This is a challenging task in particular if we restrict ourselves to a limited class of two-particle interactions by which the physical qubits can be controlled. For the special cases of four and six physical qubits we discuss a set of such quantum gates which satisfy these two conditions for the recently developed detected-jump correcting quantum codes [1]. These quantum codes are capable of stabilizing distinguishable qubits against decoherence arising from spontaneous decay processes.
In measurement-based quantum computing, there is a natural ``causal cone'' among qubits of the resource state, since the measurement angle on a qubit has to depend on previous measurement results in order to correct the effect of by-product operators. If we respect the no-signaling principle, by-product operators cannot be avoided. Here we study the possibility of acausal measurement-based quantum computing by using the process matrix framework [Oreshkov, Costa, and Brukner, Nat. Commun. 3, 1092 (2012)]. We construct a resource process matrix for acausal measurement-based quantum computing restricting local operations to projective measurements. The resource process matrix is an analog of the resource state of the standard causal measurement-based quantum computing. We find that if we restrict local operations to projective measurements the resource process matrix is (up to a normalization factor and trivial ancilla qubits) equivalent to the decorated graph state created from the graph state of the corresponding causal measurement-based quantum computing. We also show that it is possible to consider a causal game whose causal inequality is violated by acausal measurement-based quantum computing.