Quantum Computing with Cluster States Gelo Noel M. Tabia ∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 Institute for Quantum Computing and University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 15 April 2011 Summary In cluster state quantum computing, we start with a given fixed entangled state of many qubits called a cluster state and perform a quantum computation by applying a time-ordered sequence of single-qubit measurements to designated qubits in designated bases. The choice of basis for later measurements generally depend on the outcome of earlier measurements and the final result of the computation is determined from the classical information involving all measurement outcomes. This is in contrast to the more conventional network gate model of quantum computing, in which computational steps are unitary transformations. We describe how the underlying computational model of cluster state is different from the network model, in particular emphasizing how cluster state computers have no true notion of input and output registers. Nonetheless, we show that the two quantum computing models are equivalent by demonstrating how cluster states can be used to simulate quantum circuits efficiently. At the end, we include a short account on how to produce optical cluster states. ∗Email: [email protected] (UW ID: 20294659) Table of Contents 1 Introduction 1 2 What are cluster states? 1 3 Universality of cluster-state quantum computing 3 3.1 Simulating quantum circuits . 3 3.2 Random measurement outcomes . 8 3.3 Sequence of simulated quantum gates . 8 3.4 Efficiency of simulating quantum gates . 9 4 Computational model for the cluster-state quantum computer 11 4.1 Non-network character of cluster state computing . 11 4.2 A cluster-state computational model . 12 5 Optical implementation of cluster states 13 5.1 Cluster states and the Knill-Laflamme-Milburn (KLM) proposal . 13 5.2 Practical optical cluster state . 15 6 Concluding remarks 16 1 Introduction A quantum computer promises efficient information processing of some computational tasks that are believed to be intractable with classical computers. Standard quantum computing models are based on the paradigm of quantum logic networks, where a sequence of unitary quantum gates form the compu- tational circuit. The one-way quantum computer proposed by Raussendorf and Briegel [1] provides an alternative model for scalable quantum computing in which a specific form of a highly entangled multi- particle state, called a cluster state, plays the pivotal role of being a universal resource for quantum computation. To perform a quantum algorithm with a cluster state, individual qubits are measured in a temporal sequence of adaptive single-qubit measurements with classical feed-forward processing of outcomes. The classical readout for such a quantum computation essentially involves all measurement outcomes of the cluster qubits. In this project, we provide some basic details about cluster state quantum computing. Firstly, we define what cluster states are and how they can be realized from a Hamiltonian. Secondly, we show that cluster states are resources for universal quantum computing by showing how they can be imprinted with a simulated quantum circuit using single qubit rotations and CNOT gates. After that, we show how the random outcomes of individual qubit measurements do not affect the ability of a cluster state to perform an algorithm efficiently and how any necessary correction of Pauli errors due to these random outcomes can be delayed until the end of the computation. We follow that with a discussion on the non-network aspects of cluster state computing and provide a different computational model for cluster states that involves the temporal ordering of single-qubit measurements and an information flow vector that tracks the evolution of logical qubits. Lastly, we describe how cluster states can be created experimentally using polarization of photons and standard linear optical devices. 2 What are cluster states? A cluster state is a set of qubits arranged in a d-dimensional lattice at sites a 2 Zd that couple via some short-range interaction with Hamiltonian X 1 + Za 1 − Za0 H = g(t) f(a − a0) (1) int ~ 2 2 a;a0 where Zaj0ia = j0ia;Zaj1ia = −|1ia (2) defines the Pauli-Z operator. With respect to the entanglement properties of cluster states relevant to quantum computing, this Hamiltonian is equivalent to the quantum Ising model at low temperatures: X 0 HIsing = − ~g(t)f(a − a )ZaZa0 : (3) a;a0 They are equivalent in the sense that the states generated by Hint and HIsing from a given initial state are identical up to local unitary transformations applied to individual qubits. In principle, only next-neighbor interactions are needed to generate a cluster state from eq. (1). In such a case, Hint realizes simultaneous conditional phase gates between qubits at adjacent sites a and 0 a . For example, if we have a linear chain of N qubits then we can produce a cluster state using Hint 0 with f(a − a ) = δa;a0+1. 1 Suppose we have a linear chain of qubits all initially in the state j0ia + j1ia j+ia = p (4) 2 then for ' = R g(t) dt, the unitary evolution ! X 1 + Za 1 − Za+1 U(') = exp −i' (5) 2 2 a yields a disentangled chain for ' = 2(k − 1)π and a `maximally entangled' one for ' = (2k − 1)π, where k 2 Z+. For ' = π, the cluster state for a linear chain CN of N qubits can be written in compact notation as N 1 O jφiC = p (j0iaZa+1 + j1ia) (6) N N 2 a=1 where ZN+1 = 1l by definition, since it is impossible to entangle with an empty site. Examples of the smallest linear cluster states are as follows: 1 jφiC = p (j0ij+i + j1i|−i) ; (7) 2 2 1 jφiC = p (j+ij0ij+i + |−i|1i|−i) ; (8) 3 2 1 jφi = (j+ij0ij+ij0i + j+ij0i|−i|1i + |−i|1i|−i|0i + |−i|1ij+ij1i) ; (9) C4 2 which up to local unitary transformations on single qubits are equivalent to l:u: 1 jφiC = p (j0ij0i + j1ij1i) ; (10) 2 2 l:u: 1 jφiC = p (j0ij0ij0i + j1ij1ij1i) ; (11) 3 2 1 jφi l:u:= (j0ij0ij0ij0i + j0ij0ij1ij1i + j1ij1ij0ij0i − j1ij1ij1ij1i) : (12) C4 2 Observe that jφiC2 is effectively a Bell state, jφiC3 is effectively a 3-qubit GHZ state but jφiC4 is not the same as the jGHZ i = p1 (j0000i + j1111i). 4 2 So far we have only considered a linear chain of qubits. For higher-dimensional lattices (d ≥ 2), every lattice site is now specified by a vector of d integers a. Each site a has 2d neighbors. If neighboring sites are occupied by qubits, those qubits interact with the qubit in a. Let the set A specify all the lattice sites occupied by qubits. Two sites a and a0 are connected (topologically) if there exists a sequence of neighboring occupied sites, that is, N 0 fajgj=1 ; such that a1 = a; aN = a : (13) In this case, a cluster C ⊂ A is a set of connected lattice sites such that the corresponding state of the cluster qubits is given by ! O O (c+x) jφiC = j0ic Z + j1ic (14) c2C x2Γ where Γ refers to a set of steps fxg needed to move from site a to any adjacent site a0 = a + x (that 2 Figure 1: Examples of cluster states: linear, square (or rectangular), and cubic. The yellow dots represent qubits while the lines represent entanglement between nearest neighbors. is, the set ηa = fa + xj x 2 Γg (15) is the set of neighboring sites of a) and Zc+x = 1l when c+x 62 C. Some examples of clusters of different lattice dimensions is shown in fig. (1). Two important entanglement properties [2] characterizing cluster states are: (i) Maximal connectedness: Any two qubits of CN can be projected into a pure Bell state by local measurements on all the other qubits. (ii) Persistency: Pent(j i) denotes the minimum number of local measurement of qubits necessary to disentangle the state j i. It turns out that this is equivalent to the Schmidt rank, the fewest number of terms in the generalized Schmidt decomposition [3]. For cluster states, N P (jφi ) = : (16) ent CN 2 We note that a convenient, compact way to define a cluster state is via a set of eigenvalue equations KajφiC = κajφiC (17) where O Ka = Xa Za+x; κa = ±1: (18) x2Γ It turns out that in the stabilizer formalism [4, 5], the operators Ka are stabilizer generators and cluster states are their corresponding eigenstates. 3 Universality of cluster-state quantum computing 3.1 Simulating quantum circuits One can think of using the cluster state as a substrate on which quantum circuits can be imprinted on [6]. To describe this computational circuit, it is important to distinguish between the physical qubits that make up the cluster state and the logical qubits on which quantum information processing is to take place. Due to the entanglement properties of a cluster state, no individual physical qubit carries information that corresponds to an input state|rather information is contained in correlations between 3 qubits. However, for simplicity we first consider the case where an encoded input state is read into the cluster. Suppose we start with an array of qubits arranged in a square lattice and whose state is given by jφiC . Then to perform a quantum computation, we do the following: (a) Remove the subset of qubits not needed for the calculation by measuring these qubits in the Z- 0 0 basis. If the calculation uses qubits arranged in some network C , then jφiC 7! jSiCnC0 ⊗ jφ iC0 , 0 jφ iC is local unitarily equivalent to jφiC0 , which is just a smaller cluster state.