Multiple Particle Interference and Quantum Error Correction

TL;DR

This work adapts classical error-correcting codes to the quantum regime to protect information in quantum channels and computers. It develops a two-basis, coset-based error-correction framework, demonstrates simple and general codes (e.g., simplex and Hamming families) capable of correcting multiple qubits' errors, and proves that correction in two conjugate bases suffices for small subsets of errors. By deriving quantum analogs of Shannon bounds and proposing a stochastic error model, the paper shows conditions under which decoherence can be exponentially suppressed with polynomial resource overhead, suggesting the theoretical feasibility of error-free quantum computation. The results also connect quantum privacy amplification to error detection across bases and point to both practical challenges and rich future directions in quantum information theory.

Abstract

The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K/n can be greater than 1-2 H(2x/n), but must be less than 1 - 2 H(x/n). This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).

Figures (6)

• Figure 1: Encoding for simplest error correction scheme. Initially $Q$ is the qubit to be encoded, and the three 'encoding' qubits are in the state $\left|\right.\overline{000} \left.\right>$. Symbols: $\updownarrow =$ state swapping; $^| \space \times =$ controlled not in basis 2.
• Figure 2: Simplest error correction scheme. All operations take place in basis 2. After two cnot operations, the lower two qubits are measured in basis 2. The results are fed to a classical 'box' which then complements ( not operation) one or more of the qubits, depending on the measurement results. The two final cnot's reencode the state (see text).
• Figure 3: Encoder for the simplest scheme enabling single error correction in both bases. The multiple cnot symbols mean successive cnot operations carried out between the single control qubit and each of the target qubits. The initial state is $\left| {00Q0000} \right>$. The first two cnot operations prepare for the generation of a superposition of the simplex code and its complement. The rest generates the code from this preparatory state. The symbol $\bigcirc \space R$ means the rotation $\left| {0} \right> \rightarrow \left|\right.\overline{0} \left.\right>$, $\left| {1} \right> \rightarrow \left|\right.\overline{1} \left.\right>$.
• Figure 4: Error corrector for the code generated by figure \ref{['fig:enc11']}. Multiple cnot operations perform parity checks. The lower three qubits are measured, and the results used to determine which qubits undergo a not operation. The scheme is first applied in basis 1, then in basis 2 (see text).
• Figure 5: Alternative method of error correction. Codes need not be corrected 'in place' using the qubits of the computer itself (as in the previous figures). It may be more convenient to establish the error syndrome using a set of ancillary qubits. The example shown here carries out the same correction as the corrector of figure \ref{['fig:cor11']}.