

Regular Articles: New Paradigm toward Realizing Quantum Computer Reducing the Resources in Measurementonly Quantum ComputationAbstractThis article describes a recent result concerning the construction of a small set of projective measurements required for implementing universal quantum computation, that is, for implementing an arbitrary unitary operation. Projective measurements are used as the main resources for implementing an arbitrary unitary operation in the measurementonly quantum computation model in contrast to conventional models. Resource minimization is important not only for realizing a quantum computer based on the measurementonly quantum computation model, but also for understanding the computational power of projective measurements.
1. IntroductionQuantum computers are expected to enable highspeed computing and many researchers have made numerous attempts to realize them. However, a practical quantum computer has not yet been made. Most of the attempts are based on the standard model of quantum computation, which is called the quantum circuit model or gatebased quantum computation [1], [2]. The main resources for universal quantum computation, that is, for implementing an arbitrary unitary operation, are unitary gates that perform onequbit and twoqubit unitary operations. A computation proceeds by applying unitary gates to an input state to transform it into an appropriate output one. Projective measurements of the output state give us the final output of the computation, which is classical information. In 2003, on the basis of an idea described by Gottesman and Chuang [3], Nielsen proposed a model of quantum computation [4] strikingly different from the standard model. This model is called teleportationbased or measurementonly quantum computation. The main resources required for universality are only projective measurements. A computation proceeds by performing projective measurements on an input state to transform it into an appropriate output one. Projective measurements on the output state give us the final output of the computation. This model has recently attracted attention since it allows us to make completely new attempts to realize quantum computers. An important problem is to minimize the resources required for universality. From the practical standpoint, solutions to this problem will contribute to the performance of unitary operations on a quantum computer since it will be able to use only a limited amount of resources. On the theoretical side, they will contribute to our understanding of the computational power of projective measurements. Let us consider the problem of minimizing the resources required for universality in measurementonly quantum computation. There have been many studies in this direction [5]–[8]. The best known result is that a set consisting of one twoqubit projective measurement and infinitely many kinds of onequbit projective measurements with one ancillary qubit is sufficient for universal quantum computation [8]. Since it is impossible to decrease the number of twoqubit projective measurements and the number of ancillary qubits, we shall focus on onequbit projective measurements. Thus, our problem is restated as the problem of constructing a small set of onequbit projective measurements such that the set with one twoqubit projective measurement and one ancillary qubit is sufficient for universal quantum computation. The state of a qubit corresponds to a point on the unit threedimensional sphere (Fig. 1) [1] and a onequbit projective measurement corresponds to an axis, which is a line through the origin of the sphere. There are two points of intersection between an axis and the sphere. A onequbit projective measurement probabilistically projects a onequbit state into one of the two states corresponding to the two intersection points on the sphere. The best known result requires the set consisting of onequbit projective measurements corresponding to all the axes of the sphere¡Çs XY plane and the Z axis [8]. Until recently, it was not known whether a smaller set of onequbit projective measurements could be constructed for universal quantum computation.
In this article, I describe my recent result that this can be done [9]. Specifically, I show that the set consisting of onequbit projective measurements corresponding to all the axes of the sphere¡Çs XY plane (with one twoqubit projective measurement and one ancillary qubit) is sufficient for universal quantum computation. In other words, I show that the onequbit projective measurement corresponding to the Z axis can be removed from the best known set. A key finding is that the onequbit projective measurement corresponding to the Y axis can often be used in place of the onequbit projective measurements corresponding to the X and Z axes. In particular, a key ingredient of my procedure for implementing an arbitrary onequbit unitary operation is a simplified version of quantum teleportation (called state transfer) based on the onequbit projective measurement corresponding to the Y axis. 2. Measurementonly quantum computation2.1 Quantum states and projective measurementsAs described above, the state of a qubit corresponds to a point on the unit threedimensional sphere (Fig. 1). The two points of intersection of the Z axis and the sphere are represented by 0 and 1. The other points on the sphere correspond to superposition states of 0 and 1. For example, the two points of intersection of the X axis and the sphere are 0 + 1 and 0 − 1. Moreover, the two points of intersection of the Y axis and the sphere are 0 + 1 and 0 − 1. In general, a onequbit state is represented as α0 + β1, where α and β are complex numbers such that α^{2} + β^{2}=1. A onequbit projective measurement corresponds to an axis L, which is a line through the sphere¡Çs origin. It probabilistically projects a onequbit state into one of the two states corresponding to the two points of intersection of L and the sphere. The probability depends on the state being measured. The measurement gives us the classical outcome (1 or 1) representing which of the two states the original state is projected into by the measurement. We call such a measurement an Lmeasurement. For example, the Zmeasurement of a qubit in state α0 + β1 projects the state into 0 with probability α^{2} and into 1 with probability β^{2}. In general, an axis of the sphere¡Çs XY plane is represented by (cosθ)X+(sinθ)Y for some real number θ∈[0,2π). It corresponds to the onequbit projective measurement that probabilistically projects a onequbit state into one of the two states 0 + 1 and 0  1. 2.2 Measurementbased quantum circuitsA computational procedure in measurementonly quantum computation can be represented by a measurementbased quantum circuit (very similar to the standard quantum circuit [2]). It consists of wires and measurement gates that correspond to qubits and projective measurements, respectively. In a circuit diagram, a wire is represented by a horizontal line and a measurement gate is represented by the axis symbol corresponding to the projective measurement on the wires on which it is performed. Information flows through the circuit from left to right. As an example, consider the measurementbased quantum circuit depicted in Fig. 2. It represents the following procedure, where the initial state of qubit 1 is φ and that of qubit 2 is an arbitrary onequbit state. (1) Perform the Xmeasurement on qubit 2. (2) Perform the ZZmeasurement on qubits 1 and 2. (3) Perform the Xmeasurement on qubit 1. I do not give details of the ZZmeasurement here, but in general it generates a twoqubit state that cannot be represented as a product form (called entangled state).
The procedure outputs the state σ φ on qubit 2 for some unitary operation σ, which is in a special class of unitary operations called Pauli operations. An important point is that σ is determined by the classical outcomes of the projective measurements in the procedure, and the inverse of s can be performed by the Ymeasurements (and one twoqubit projective measurement and one ancillary qubit). Thus, σ can be easily removed and thus ignored. That is, the procedure transfers an arbitrary onequbit state from qubit 1 to qubit 2 (up to Pauli operations) and thus is called state transfer [6]. It can be regarded as a simplified version of quantum teleportation. As shown in [6], state transfer is a key ingredient of a procedure for implementing an arbitrary onequbit unitary operation. More precisely, a procedure for implementing such an operation (up to Pauli operations) can be obtained by replacing the projective measurements in the state transfer with ones that depend on the desired operation. For example, a procedure for implementing a Hadamard operation H can be obtained by replacing the first measurement with the Zmeasurement and the second measurement with the ZXmeasurement, where H is the onequbit unitary operation that maps 0 and 1 to 0 + 1 and 0 − 1, respectively. 3. Universal set of projective measurementsI will show that the set consisting of the onequbit projective measurement corresponding to the axis (cosθ)X+(sinθ)Y for any θ∈[0,2π) (with one twoqubit projective measurement and one ancillary qubit) is sufficient for universal quantum computation. To do this, I will show that an arbitrary unitary operation can be implemented using only the above projective measurements. Since an arbitrary unitary operation can be implemented by combining onequbit unitary operations with a twoqubit unitary operation [1], it suffices to show how to implement these operations. 3.1 New state transferA key ingredient of my procedure for implementing an arbitrary onequbit unitary operation is a new state transfer based on the Ymeasurements. Consider the following procedure (Fig. 3), where the initial state of qubit 1 is φ and that of qubit 2 is an arbitrary onequbit state. (1) Perform the Ymeasurement on qubit 2. (2) Perform the ZZmeasurement on qubits 1 and 2. (3) Perform the Ymeasurement on qubit 1.
The procedure can be shown to be a state transfer, which transfers the input state from qubit 1 to qubit 2. It is obtained by replacing the Xmeasurements in the previous state transfer with the Ymeasurements. That is, this is an example of the case where the Ymeasurements can be used in place of the Xmeasurements. 3.2 Implementations of onequbit and twoqubit unitary operationsI will deal with onequbit unitary operations first. I can show that, by replacing the ZZmeasurement in the new state transfer with the ZXmeasurement, the resulting procedure is the one for implementing H. An important point is that it uses only Ymeasurements, though the previous procedure for implementing H uses the X and Zmeasurements as described above. This can be considered an example of the case where the Ymeasurements can be used in place of the X and Zmeasurements. That is, the new state transfer provides an H implementation procedure that requires only a small set of onequbit projective measurements. By generalizing the method for transforming the new state transfer into an H implementation procedure, I can obtain a procedure for implementing an arbitrary onequbit unitary operation (up to Pauli operations). Moreover, I can show that it requires only the onequbit projective measurement corresponding to the axis (cosθ)X+(sinθ)Y for any θ∈[0,2π). As a twoqubit unitary operation, it suffices to deal with the controlled Z operation Λ Z that maps 00, 01, 10, and 11 to 00, 01, 10, and −11, respectively. The remaining problem is to obtain a procedure for implementing Λ Z (up to Pauli operations) using only the onequbit projective measurement corresponding to the axis (cosθ)X+(sinθ)Y for any θ∈[0,2π). Though it is difficult to implement Λ Z directly, I can show that there is a twoqubit unitary operation similar to Λ Z such that combining it with onequbit unitary operations implements an arbitrary unitary operation and that it can be implemented by using only Ymeasurements. Thus, the set consisting of the onequbit projective measurement corresponding to the axis (cosθ)X+(sinθ)Y for any θ∈[0,2π) is sufficient for universal quantum computation. 4. Conclusions and future workI examined the problem of minimizing the resources required for universality in measurementonly quantum computation and described a small set of projective measurements sufficient for universal quantum computation. A key ingredient of my procedures is state transfer based on the Ymeasurements. It would be interesting to consider approximate universality in measurementonly quantum computation [10] because a small approximately universal set of projective measurements is particularly important in practice. Moreover, it would be interesting to investigate the resources required for other important problems, such as graph state preparation [9]. References
