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arXiv:quant-ph/9702033 · Physical Review A

Five quantum register error correction code for higher spin systems

H. F. Chau

Published Jul 1, 1997• 42 citations
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Abstract

I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each $N$-state quantum particle is encoded as five $N$-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.

DOI
10.1103/physreva.56.r1

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I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each NNN-state quantum particle is encoded as five NNN-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.
I consider the theory of the quantum error correcting code (QECC), where each quantum particle has more than two possible eigenstates. In this higher spin system, I report an explicit QECC that is related to the symmetry group ${\mathrm{Z}}_{2}^{\ensuremath{\bigotimes}(\mathrm{N}\mathrm{\ensuremath{-}}1)}$\ensuremath{\bigotimes}SN{\mathrm{S}}_{\mathrm{N}}SN​. This QECC, which generalizes Shor's simple majority vote code [Phys. Rev. A 52, 2493 (1995)], is able to correct errors arising from exactly one quantum particle. I also provide a simple encoding algorithm.
Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correction code (QECC), with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this notoriously difficult challenge, we push the limits of the depth of superconducting quantum circuits and experimentally realise the universal five-qubit QECC, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the five-qubit QECC for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of 57.1(3)%57.1(3)\%57.1(3)% while with a high fidelity of 98.6(1)%98.6(1)\%98.6(1)% in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of 97.2(2)%97.2(2)\%97.2(2)% within the code space. Finally, we realise the decoding circuit and recover the input state with a process fidelity of 57.4(7)%57.4(7)\%57.4(7)%. After decoding, by identifying errors via the measurement of the four ancillae and then mathematically recovering the qubit state, we verify the power of error correction of this code. Thus, by demonstrating each key aspect of error correction with the five-qubit code, our work establishes the viability of experimental quantum error correction with superconducting qubits and paves the route to fault-tolerant quantum computing.
We present a quantum error correction code which protects a qubit of information against general one qubit errors. To accomplish this, we encode the original state by distributing quantum information over five qubits, the minimal number required for this task. We describe a circuit which takes the initial state with four extra qubits in the state ∣0〉|0〉∣0〉 to the encoded state. It can also be converted into a decoder by running it backward. The original state of the encoded qubit can then be restored by a simple unitary transformation.
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.
We present a quantum error correction code which protects a qubit of information against general one qubit errors which maybe caused by the interaction with the environment. To accomplish this, we encode the original state by distributing quantum information over five qubits, the minimal number required for this task. We give a simple circuit which takes the initial state with four extra qubits in the state |0> to the encoded state. The circuit can be converted into a decoding one by simply running it backward. Reading the extra four qubits at the decoder's output we learn which one of the sixteen alternatives (no error plus all fifteen possible 1-bit errors) was realized. The original state of the encoded qubit can then be restored by a simple unitary transformation.