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Cluster State Quantum Computing 1 Cluster State Quantum Computing CIS:410/510 Final Report, Spring 2016 Dileep V. Reddy, Mayra Amezcua, Zach Schmidt [email protected], [email protected], [email protected] Abstract— Any quantum computation can be performed via se- A. Preparation of linear cluster state quences of one-qubit measurements on a specific type of initially entan- gled state – the cluster state. Each computational step is a projective Intuitively, a cluster state can be thought of as a graph measurement that destroys a quantum state, leaving a final state that where every vertex represents a qubit, and every edge rep- relies on the outcomes of earlier computations. The model of interest resents the application of a Cz gate to both adjacent vertices. is the one-way quantum computer which is based on this measurement scheme. This paper will present background regarding computation us- ing only measurements, a brief introduction into the preparation of clus- ter states, a discussion of one way quantum computers (1WQC), and the computational power of various configurations of a 1WQC, and will end with an overview of physical implementations. I. BACKGROUND Over the past few decades, advances in science and tech- nology have greatly contributed to the development of mod- ern computers. While these computers are efficient and con- venient for everyday needs, they fail at certain computational tasks. Instead, quantum computers promise faster large scale factorization and database searches that are intractable for their classical counterparts. The first quantum computer de- signs were based off of classical models; sequences of one- and multi-qubit gate operations are performed on chosen Fig. 1. Figure from [3], showing representative 2D cluster shapes. The ver- quantum bits and a final measurement would convert quan- tices are qubits with integer indices, and the edges indicate entanglement connectivity between select neighbors. tum information into classical bits. A new model, proposed by Briegel and Raussendorf [1], demonstrates that quantum A cluster state can be represented as a graph G = (N; E), computation can be achieved by using single qubit measure- where the n 2 N is a qubit and e 2 E is the application of a ments as computational steps. This so-called cluster model Controlled-Z (Cz) gate, where: or one-way quantum computer (1WQC) relies on an entan- gled state of a large number of qubits or cluster state as the 01 0 0 0 1 resource. The fascinating feature about 1WQC is that they B0 1 0 0 C Cz = B C have no classical analogues and probe into new territory in @0 0 1 0 A regards to entanglement and measurements. 0 0 0 −1 II. CLUSTER STATES A linear cluster state is one where degree(n) ≤ 28n 2 N. Consider a set of qubits C labeled by an integer index, that are distributed in some lattice such that every qubit can be said to have adjacent neighbors. For these to collectively 1 2 3 4 form a cluster state, their quantum mechanical state would 4-Node Linear Cluster State be characterized by the set of eigenvalue equations [2], A method to prepare such a cluster state is given in [4], Ka jΦiC = κ jΦiC (1) consisting of “cascading” Cz gates on n qubits as follows: for a family of operators K = X(a) N Z(γ), a 2 C, 1 2 3 a γ2Γ(a) j+i • where Γ(a) is the set of indices of all qubits in the “adjacent 1 (a) neighborhood” of a. The matrix X is used to denote an j+i2 CZ • X operation on qubit-a, and so on. The eigenvalue κ = ±1 j+i CZ • is determined by the specific occupation pattern of the neigh- 3 boring sites. j+i4 CZ 2 We can then analyze the state of the qubits at each of the It is important to emphasize that the order in which the Cz dotted lines: gates are applied to grow the cluster state is irrelevant, as all of these pair-wise operations commute. This feature will be 1: exploited later when discussing parallelizability. j0i1 j+i2 + j1i1 j−i2 p j+i3 j+i4 2 III. THE EFFECTS OF MEASUREMENT ON A CLUSTER j+i j0i + j−i j1i ≡ 1 2p 1 2 j+i j+i STATE 2 3 4 As is clear from the form of the expressions of all cluster 2: states illustrated thus far, measuring any node in the com- j+i1 j0i2 j+i3 + j−i1 j1i2 j−i3 putational basis severs it from the remaining graph by cut- p j+i4 2 ting all of it’s edges with it’s neighboring nodes. Should the j0i1 j+i2 + j1i1 j−i2 j0i1 j−i2 + j1i1 j+i2 outcome of said measurement be 1, then a Z gate/transform ≡ j0i3 + j1i3 j+i4 2 2 gets applied to all of it’s erstwhile neighbors in the leftover 3: cluster state. Thus, a large cluster state can be arbitrarily j+i j0i j+i + j−i j1i j−i trimmed, split, and/or reshaped by removing qubits from the 1 2 3 1 2 3 j0i 2 4 cluster. This is accomplished by measuring the target qubit in j+i j0i j−i + j−i j1i j+i 1 2 3 1 2 3 the computational basis, and performing appropriate unitary + j1i4 2 rotations on its former neighbors based on the measurement (j+i j0i + j−i j1i ) j0i j+i + (j+i j0i − j−i j1i ) j1i j−i ≡ 1 2 1 2 3 4 1 2 1 2 3 4 outcome. 2 The effect of an X-measurement (i.e., a computational ba- The action of the C gate in the computational basis can z sis measurement following a Hadamard transformation) on be seen to be jx; yi ! (−1)xy jx; yi. Cluster states of ar- any node of the cluster state is much more involved. This is bitrary shape and connectivity can similarly be prepared via best illustrated when demonstrating the use of a linear clus- the recursive use of the Hadamard gate and two-qubit fusion ter state as a wire for quantum information. For this exer- operations [5], [3]. cise, we start with a linear cluster state with three nodes (la- B. Preparation of T-shaped cluster state beled 1, 2, and 3). A single qubit of quantum information j i = α j0i + β j1i is stored in a physical qubit labeled 0 as A cluster state without the limitation on the degree of a illustrated below. node allows us to build nonlinear cluster states: 1 2 3 j i0 0 1 2 3 4 Gate C(0;1), followed by measurements M (0), M (1),& M (2). 4-Node T-Shaped Cluster State z X X X The circuit creating this cluster state will look as follows: To teleport the state j i to physical qubit number 3, we must first supply the quantum information to the “wire.” This j+i • 1 is achieved by applying a Cz gate between physical qubits 0 and 1. Using jLCi to denote the linear cluster state, we j+i2 CZ • • 123 have j+i3 CZ j+i4 CZ (0;1) Cz j i0 ⊗ jLCi123 The state of the qubits after the application of the first two 1 = p [α j0i j+i j0i j+i + β j1i j−i j0i j+i Cz gates is identical to the linear case, and the state after the 2 0 1 2 3 0 1 2 3 last Cz (at the dotted line) is given by: α j0i0 j−i1 j1i2 j−i3 + β j1i0 j+i1 j1i2 j−i3] j+i j0i j+i j+i + j−i j1i j−i j−i Following this, we perform X-measurements on physi- 1 2 3 4p 1 2 3 4 2 cal qubits 0, 1, and 2 in that order. Let us denote an X- measurement operation on the jth-node with M (j), and let 1 j+i1 j0i2 j+i3 + j−i1 j1i2 j−i3 X ≡ p p j0i the outcome of any measurement on the same node be m . 2 2 4 j Then, the end result of these operations is the state j+i1 j0i2 j+i3 − j−i1 j1i2 j−i3 + p j1i4 2 m2 m1 m0 X Z X H j i3 ; mj 2 f0; 1g: 3 The quantum information j i has successfully been tele- ray of entangled qubits, information propagates horizontally ported to physical qubit 3, up to application of Pauli opera- through a row of qubits while vertical qubit neighbors are tors depending on the measurement outcomes. Since the left- used for two-qubit gates. Similarly, three-dimensional clus- over/extra Pauli operators do not commute, these measure- ters can be used to implement topologically protected gates ments had to have been carried out in a specific order. How- [7], where the gate function only depends upon the way “con- (2;3) ever, the operation Cz , which was employed to grow the nected defects” are wound around one another, but not on the (0;1) details of their shape. This degree of freedom affords the de- linear cluster state, commutes with Cz , as well as mea- (0=1) sign some fault tolerance. surements MX . Thus, further links to the chain can be grown as earlier links are being subjected to measurements. The basic principle of cluster-state quantum computa- This aids in parallelizability, as well as physical implementa- tion is to effectively enact arbitrary quantum circuits onto tion of cluster state quantum computing schemes.
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