arXiv:1001.1295v1 [quant-ph] 8 Jan 2010 Low-temperature coherence properties of Z$_{2}$ quantum memory Tomoyuki Morimae ∗ Department of Basic Science, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8902, Japan and Laboratoire Paul Painlev´ e, Universit´ e Lille 1, 59655 Villeneuve d’Ascq C´ edex, France (Dated: November 16, 2018) We investigate low-temperature coherence properties of the Z 2 quantum memory which is capable of storing the information of a single logical qubit. We show that the memory has superposition of macroscopically distinct states for some values of a control parameter and at sufficiently low temperature, and that the code states of this memory have no instability except for the inevitable one. However, we also see that the coherence power of this memory is limited by space and time. We also briefly discuss the RVB memory, which is an improvement of the Z 2 quantum memory, and the relations of our results to the obscured symmetry breaking in statistical physics. PACS numbers: 03.65.Aa, 03.67.-a, 03.65.Ta, 05.30.-d Although Kitaev’s toric code is highly sophisticated and indeed has inspired plenty of successive studies [2, 6–13], it is not easy to implement a scalable toric code in a laboratory since non-local operations are required. A complementary approach is therefore also important from the practical point of view. In this paper, we study the less elaborate but more fea- sible quantum memory, namely the Z$_{2}$ quantum memory. Although the Z$_{2}$ quantum memory is too primitive to be a complete and universal quantum memory, it is still valuable to study it, since the simple structure of this memory means feasibility in a laboratory and the pos- sibility of capturing the essence of theoretical aspects of quantum memory. In general, a quantum memory must have a large coherence for some values of a control pa- rameter in order to store a coherent information of logical qubits. Therefore, we analyze the coherent properties of the Z$_{2}$ quantum memory at low temperature, by using the method of detecting superposition of macroscopically distinct states developed in Refs. [5, 14–16]. We show that the Z$_{2}$ quantum memory can have superposition of macroscopically distinct states for some values of a con- trol parameter and at sufficiently low temperature, and that the code states have no instability except for the inevitable one. However, we also see that the power of superposition of macroscopically distinct states in this memory is limited by space and time. These results sug- gest that the Z$_{2}$ quantum memory is of limited use as a prototype of a small quantum memory. Quantum memory [1, 2], which stores the coherent in- formation of logical qubits, is an essential ingredient of quantum information processings, and plays crucial roles in almost all fields of quantum information science [3]. Basically, a quantum memory consists of macroscopically many physical qubits which encode the state of few log- ical qubits. Each logical basis should be encoded on macroscopically distinct physical states since otherwise the indistinguishability of logical bases is easily destroyed by local errors. On the other hand, in order to store quantum coherent superposition of these logical bases, the memory must be able to have superposition of macro- scopically distinct states of physical qubits. To maintain such superposition is very difficult. First, if the size of the memory is infinite, such macroscopic superposition is impossible since superposition of different phases is just a classical mixture of them in an infinite system [4–6]. Second, even in finite systems, such macroscopic super- position is usually very unstable [5]. Therefore, how to balance those two contradictory demands (i.e., classical information must be encoded on macroscopically distinct states but we also need superposition of them) is one of the most challenging problem in the implementation of quantum memory. The code space of a quantum memory is often realized as the lowest-energy eigenspace of a many-body Hamil- tonian. One of the most beautiful examples is Kitaev’s toric code [1]. Kitaev introduced a four-body Hamilto- nian which exhibits a topological phase transition. The ground states are degenerated and energetically isolated from the excited states. Each of these ground states cor- responds to different topological phases in the thermo- dynamic limit, and therefore logical bases encoded on these degenerate ground states are immune to local er- rors. Logical qubit operations are performed by applying long strings of Pauli operators on physical qubits, which means that logical operations are non-local. This paper is organized as follows. In the next sec- tion, we briefly review the method of detecting super- position of macroscopically distinct states. In Sec. III, we study the zero-temperature case. We next study the finite-temperature case in Sec. IV. Finally, in Sec. V, we briefly discuss the RVB quantum memory, which is an improvement of the Z$_{2}$ quantum memory, and the rela- tions of our results to the concept of symmetry breaking in statistical physics in Sec. VI. ∗ morimae@asone.c.u-tokyo.ac.jp I. INTRODUCTION