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arXiv:1301.4796 · Physical Review A

Strategy for implementing stabilizer-based codes on solid-state qubits

Tetsufumi Tanamoto, Vladimir M. Stojanović, Christoph Bruder, Daniel Becker

Published May 6, 2013• 12 citations
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Abstract

We present a method for implementing stabilizer-based codes with encoding schemes of the operator quantum error correction paradigm, e.g., the ``standard'' five-qubit and CSS codes, on solid-state qubits with Ising or $XY$-type interactions. Using pulse sequences, we show how to dynamically generate the effective dynamics of the stabilizer Hamiltonian, the sum of an appropriate set of stabilizer operators for a given code. Within this approach, the encoded states (ground states of the stabilizer Hamiltonian) can be prepared without measurements and preserved against both the time evolution governed by the original qubit Hamiltonian, and errors caused by local sources.

DOI
10.1103/physreva.87.052305

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We present a method for implementing stabilizer-based codes with encoding schemes of the operator quantum error correction paradigm, e.g., the ``standard'' five-qubit and CSS codes, on solid-state qubits with Ising or XYXYXY-type interactions. Using pulse sequences, we show how to dynamically generate the effective dynamics of the stabilizer Hamiltonian, the sum of an appropriate set of stabilizer operators for a given code. Within this approach, the encoded states (ground states of the stabilizer Hamiltonian) can be prepared without measurements and preserved against both the time evolution governed by the original qubit Hamiltonian, and errors caused by local sources.
In quantum error-correcting code (QECC), many quantum operations and measurements are necessary to correct errors in logical qubits. In the stabilizer formalism, which is widely used in QECC, generators Gi(i=1,2,...){G}_{i}\phantom{\rule{0.28em}{0ex}}(i=1,2,...)Gi​(i=1,2,...) consist of multiples of Pauli matrices and perform encoding, decoding, and measurement. In order to maintain encoding states, the stabilizer Hamiltonian ${H}_{\mathrm{stab}}=\ensuremath{-}{\ensuremath{\sum}}_{i}{G}_{i}$ is suitable because its ground state corresponds to the code space. On the other hand, Hamiltonians of most solid-state qubits have two-body interactions and show their own dynamics. In addition solid-state qubits are fixed on substrate and qubit-qubit operation is restricted in their neighborhood. The main purpose of this paper is to show how to directly generate the stabilizer Hamiltonian Hstab{H}_{\mathrm{stab}}Hstab​ from conventional two-body Hamiltonians with Ising interaction and XYXYXY interaction by applying a pulse control method such as an NMR technique. We show that generation times of Hstab{H}_{\mathrm{stab}}Hstab​ for nine-qubit code, five-qubit code, and Steane code are estimated to be less than 300 ns when typical experimental data of superconducting qubits are used, and sufficient pulse control is assumed. We also show how to prepare encoded states from an initial state $|0,...,0\ensuremath{\rangle}$. In addition, we discuss an appropriate arrangement of two- or three-dimensional arrayed qubits.
In this paper we investigate stabilizer quantum error correction codes using controlled phase rotations of strong coherent probe states. We explicitly describe two methods to measure the Pauli operators that generate the stabilizer group of a quantum code. First, we show how to measure a Pauli operator acting on physical qubits using a single coherent state with large average photon number, displacement operations, and photon detection. Second, we show how to measure the stabilizer operators fault-tolerantly by the deterministic preparation of coherent quantum superposition (``cat'') states along with one-bit teleportations between a qubitlike encoding of coherent states and physical qubits.
Proposals for quantum computing devices are many and varied. They each have unique noise processes that make none of them fully reliable at this time. There are several error correction/avoidance techniques which are valuable for reducing or eliminating errors, but not one, alone, will serve as a panacea. One must therefore take advantage of the strength of each of these techniques so that we may extend the coherence times of the quantum systems and create more reliable computing devices. To this end we give a general strategy for using dynamical decoupling operations on encoded subspaces. These encodings may be of any form; of particular importance are decoherence-free subspaces and quantum error correction codes. We then give means for empirically determining an appropriate set of dynamical decoupling operations for a given experiment. Using these techniques, we then propose a comprehensive encoding solution to many of the problems of quantum computing proposals which use exchange-type interactions. This uses a decoherence-free subspace and an efficient set of dynamical decoupling operations. It also addresses the problem of controllability in solid-state quantum dot devices.
In this paper, we deal with quantum information decoding. Quantum codes are a known technique for correcting errors of quantum information caused by noise or operational errors and one of the most popular types of codes is the so-called stabilizer code. Various implementation methods have been proposed to encode information into a stabilizer code, but there have been few studies on decoding to extract the original information from the code words. In this paper, we propose a decoding method for stabilizer codes using dissipation dynamics and show that the original information can be extracted from the code words.
Protecting quantum information from the detrimental effects of decoherence and lack of precise quantum control is a central challenge that must be overcome if a large robust quantum computer is to be constructed. The traditional approach to achieving this is via active quantum error correction using fault-tolerant techniques. An alternative to this approach is to engineer strongly interacting many-body quantum systems that enact the quantum error correction via the natural dynamics of these systems. Here we present a method for achieving this based on the concept of concatenated quantum error correcting codes. We define a class of Hamiltonians whose ground states are concatenated quantum codes and whose energy landscape naturally causes quantum error correction. We analyze these Hamiltonians for robustness and suggest methods for implementing these highly unnatural Hamiltonians.
Stabilizer is a very important description method of quantum error-correcting code. Before to generate quantum codes based on the stabilizer it needs to generate seed generator. The problem of finding seed generators is transformed into the problems of determining linear independence or linear dependence of vectors and finding the solutions of equations; we also present quick determinant method, sort and find method and solve equations method to verify the validity of the operator; we present the total number of seed generators of the stabilizer code. The method introduced in this paper can be applied to the quantum stabilizer code contains both type 1 and type 2 generator; and also can be applied to the quantum stabilizer code contains only type 1 generators. Through standardize the generator of the stabilizer and the seed generators at the same time; it can quickly get the quantum encoding circuit. It also presented the number of 2-qubit gates and 1-qubit gates needed for the encoding circuit.