Stabilizer States as a Basis for Density Matrices

arXiv:1112.2156v1 [quant-ph] 9 Dec 2011 Stabilizer States as a Basis for Density Matrices Simon J. Gay School of Computing Science, University of Glasgow, UK August 28, 2018 Abstract We show that the space of density matrices for n-qubit states, considered as a (2n)2-dimensional real vector space, has a basis consisting of density matrices of sta- bilizer states. We describe an application of this result to automated verification of quantum protocols. 1 Definitions and Results We are working with the stabilizer formalism [5], in which certain quantum states on sets of qubits are repre- sented by the intersection of their stabilizer groups with the group generated by the Pauli operators. The sta- bilizer formalism is defined, explained and illustrated in a substantial literature; good introductions are given by Aaronson and Gottesman [1] and Nielsen and Chuang [7, Sec. 10.5]. In this paper we only need to use the following facts about stabilizer states. 1. The standard basis states are stabilizer states. 2. The set of stabilizer states is closed under applica- tion of Hadamard (H), Pauli (X, Y, Z), controlled not (CNot), and phase (P = 1 0 0 i  ) gates. 3. The set of stabilizer states is closed under tensor product. Notation 1 Write the standard basis for n-qubit states as {|x⟩| 0 ⩽x < 2n}, considering numbers to stand for their binary representations. We omit normalization factors when writing quantum states. Definition 1 Let GHZn = |0⟩+ |2n −1⟩and iGHZn = |0⟩+ i|2n −1⟩, as n-qubit states. Lemma 1 For all n, GHZn and iGHZn are stabilizer states. Proof: By induction on n. For the base case (n = 1), we have that |0⟩+ |1⟩and |0⟩+ i|1⟩are stabilizer states, by applying H and then P to |0⟩. For the inductive case, GHZn and iGHZn are obtained from GHZn−1 ⊗|0⟩and iGHZn−1 ⊗|0⟩, respectively, by applying CNot to the two rightmost qubits. □ Lemma 2 If 0 ⩽x, y < 2n and x ̸= y then |x⟩+|y⟩and |x⟩+ i|y⟩are stabilizer states. Proof: By induction on n. For the base case (n = 1), the closure properties imply that |0⟩+ |1⟩, |0⟩+ i|1⟩and |1⟩+ i|0⟩= |0⟩−i|1⟩are stabilizer states. For the inductive case, consider the binary represen- tations of x and y. If there is a bit position in which x and y have the same value b, then |x⟩+ |y⟩is the tensor product of |b⟩with an (n −1)-qubit state of the form |x′⟩+ |y′⟩, where x′ ̸= y′. By the induction hypothesis, |x′⟩+ |y′⟩is a stabilizer state, and the conclusion follows from the closure properties. Similarly for |x⟩+ i|y⟩. Otherwise, the binary representations of x and y are complementary bit patterns. In this case, |x⟩+ |y⟩can be obtained from GHZn by applying X to certain qubits. The conclusion follows from Lemma 1 and the closure properties. The same argument applies to |x⟩+ i|y⟩, using iGHZn. □ Theorem 1 The space of density matrices for n-qubit states, considered as a (2n)2-dimensional real vector space, has a basis consisting of density matrices of n- qubit stabilizer states. Proof: This is the space of Hermitian matrices and its obvious basis is the union of {|x⟩⟨x| | 0 ⩽x < 2n} (1) {|x⟩⟨y| + |y⟩⟨x| | 0 ⩽x < y < 2n} (2) {−i|x⟩⟨y| + i|y⟩⟨x| | 0 ⩽x < y < 2n}. (3) Now consider the union of {|x⟩⟨x| | 0 ⩽x < 2n} (4) {(|x⟩+ |y⟩)(⟨x| + ⟨y|) | 0 ⩽x < y < 2n} (5) {(|x⟩+ i|y⟩)(⟨x| −i⟨y|) | 0 ⩽x < y < 2n}. (6) This is also a set of (2n)2 states, and it spans the space because we can obtain states of forms (2) and (3) by subtracting states of form (4) from those of forms (5) and (6). Therefore it is a basis, and by Lemma 2 it consists of stabilizer states. □ 1 2 Applications The stabilizer formalism can be used to implement an efficient classical simulation of quantum computation, if quantum operations are restricted to those under which the set of stabilizer states is closed — i.e. the Clifford group operations. This result is the Gottesman-Knill Theorem [6]. Gay, Nagarajan and Papanikolaou [3, 4, 8] have applied it to formal verification of quantum sys- tems, adapting model-checking techniques [2] from clas- sical computer science. A simple example of model- checking is the following. Consider a quantum teleportation protocol as a sys- tem with one qubit as input and one (different) qubit as output; call this system Teleport. Also consider the one-qubit identity operator I. Then the specification that teleportation should satisfy is that Teleport = I, where equality means the same transformation of a one- qubit state. The aim of model-checking in this context is to automatically verify that this specification is satis- fied, by simulating the operation of the two systems on all possible inputs. For this to be possible, the telepor- tation protocol is first expressed in a formal modelling language analogous to a programming language. The approach of Gay, Nagarajan and Papanikolaou re- duces the problem to that of simulating teleportation on stabilizer states as inputs, which can be done efficiently because the teleportation protocol itself only uses Clif- ford group operations. Correctness of telepo

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