Quantum error correction

The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. This stops being the case for a quantum channel in which, due to the no-cloning theorem, it is no longer possible to repeat a single qubit three times. To overcome this, a different method first proposed by Asher Peres in 1985 [1] , such as the so-called three-qubit bit flip code, has to be used. This technique uses entanglement and syndrome measurements and is comparable in performance with the repetition code.

Quantum circuit of the bit flip code

Consider the situation in which we want to transmit the state of a single qubit ${\displaystyle \vert \psi \rangle }$ through a noisy channel ${\displaystyle {\mathcal {E}}}$. Let us moreover assume that this channel either flips the state of the qubit, with probability ${\displaystyle p}$, or leaves it unchanged. The action of ${\displaystyle {\mathcal {E}}}$ on a general input ${\displaystyle \rho }$ can therefore be written as ${\displaystyle {\mathcal {E}}(\rho )=(1-p)\rho +p\ X\rho X}$.

Let ${\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }$ be the quantum state to be transmitted. With no error correcting protocol in place, the transmitted state will be correctly transmitted with probability ${\displaystyle 1-p}$. We can however improve on this number by encoding the state into a greater number of qubits, in such a way that errors in the corresponding logical qubits can be detected and corrected. In the case of the simple three-qubit repetition code, the encoding consists in the mappings ${\displaystyle \vert 0\rangle \rightarrow \vert 0_{L}\rangle \equiv \vert 000\rangle }$ and ${\displaystyle \vert 1\rangle \rightarrow \vert 1_{L}\rangle \equiv \vert 111\rangle }$. The input state ${\displaystyle \vert \psi \rangle }$ is encoded into the state ${\displaystyle \vert \psi '\rangle =\alpha _{0}\vert 000\rangle +\alpha _{1}\vert 111\rangle }$. This mapping can be realized for example using two CNOT gates, entangling the system with two ancillary qubits initialized in the state ${\displaystyle \vert 0\rangle }$.[2] The encoded state ${\displaystyle \vert \psi '\rangle }$ is what is now passed through the noisy channel.

The channel acts on ${\displaystyle \vert \psi '\rangle }$ by flipping some subset (possibly empty) of its qubits. No qubit is flipped with probability ${\displaystyle (1-p)^{3}}$, a single qubit is flipped with probability ${\displaystyle 3p(1-p)^{2}}$, two qubits are flipped with probability ${\displaystyle 3p^{2}(1-p)}$, and all three qubits are flipped with probability ${\displaystyle p^{3}}$. Note that a further assumption about the channel is made here: we assume that ${\displaystyle {\mathcal {E}}}$ acts equally and independently on each of the three qubits in which the state is now encoded. The problem is now how to detect and correct such errors, without at the same time corrupting the transmitted state.

Comparison of output minimum fidelities, with (red) and without (blue) error correcting via the three qubit bit flip code. Notice how, for ${\displaystyle p\leq 1/2}$, the error correction scheme improves the fidelity.

Let us assume for simplicity that ${\displaystyle p}$ is small enough that the probability of more than a single qubit being flipped is negligible. One can then detect whether a qubit was flipped, without also querying for the values being transmitted, by asking whether one of the qubits differs from the others. This amounts to performing a measurement with four different outcomes, corresponding to the following four projective measurements:

$${\displaystyle {\begin{aligned}P_{0}&=|000\rangle \langle 000|+|111\rangle \langle 111|,\\P_{1}&=|100\rangle \langle 100|+|011\rangle \langle 011|,\\P_{2}&=|010\rangle \langle 010|+|101\rangle \langle 101|,\\P_{3}&=|001\rangle \langle 001|+|110\rangle \langle 110|.\end{aligned}}}$$

This can be achieved for example by measuring ${\displaystyle Z_{1}Z_{2}}$ and then ${\displaystyle Z_{2}Z_{3}}$. This reveals which qubits are different from which others, without at the same time giving information about the state of the qubits themselves. If the outcome corresponding to ${\displaystyle P_{0}}$ is obtained, no correction is applied, while if the outcome corresponding to ${\displaystyle P_{i}}$ is observed, then the Pauli X gate is applied to the ${\displaystyle i}$-th qubit. Formally, this correcting procedure corresponds to the application of the following map to the output of the channel:

$${\displaystyle {\mathcal {E}}_{\operatorname {corr} }(\rho )=P_{0}\rho P_{0}+\sum _{i=1}^{3}X_{i}P_{i}\rho \,P_{i}X_{i}.}$$

Note that, while this procedure perfectly corrects the output when zero or one flips are introduced by the channel, if more than one qubit is flipped then the output is not properly corrected. For example, if the first and second qubits are flipped, then the syndrome measurement gives the outcome ${\displaystyle P_{3}}$, and the third qubit is flipped, instead of the first two. To assess the performance of this error correcting scheme for a general input we can study the fidelity ${\displaystyle F(\psi ')}$ between the input ${\displaystyle \vert \psi '\rangle }$ and the output ${\displaystyle \rho _{\operatorname {out} }\equiv {\mathcal {E}}_{\operatorname {corr} }({\mathcal {E}}(\vert \psi '\rangle \langle \psi '\vert ))}$. Being the output state ${\displaystyle \rho _{\operatorname {out} }}$ correct when no more than one qubit is flipped, which happens with probability ${\displaystyle (1-p)^{3}+3p(1-p)^{2}}$, we can write it as ${\displaystyle [(1-p)^{3}+3p(1-p)^{2}]\,\vert \psi '\rangle \langle \psi '\vert +(...)}$, where the dots denote components of ${\displaystyle \rho _{\operatorname {out} }}$ resulting from errors not properly corrected by the protocol. It follows that

$${\displaystyle F(\psi ')=\langle \psi '\vert \rho _{\operatorname {out} }\vert \psi '\rangle \geq (1-p)^{3}+3p(1-p)^{2}=1-3p^{2}+2p^{3}.}$$

This fidelity is to be compared with the corresponding fidelity obtained when no error correcting protocol is used, which was shown before to equal ${\displaystyle {1-p}}$. A little algebra then shows that the fidelity after error correction is greater than the one without for ${\displaystyle p<1/2}$. Note that this is consistent with the working assumption that was made while deriving the protocol (of ${\displaystyle p}$ being small enough).

Quantum circuit of the phase flip code

Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with quantum computers, the sign flip. Through the transmission in a channel the relative sign between ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$ can become inverted. For instance, a qubit in the state ${\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}}$ may have its sign flip to ${\displaystyle |+\rangle =(|0\rangle +|1\rangle )/{\sqrt {2}}.}$

The original state of the qubit

${\displaystyle |\psi \rangle =\alpha _{0}|+\rangle +\alpha _{1}|-\rangle }$

will be changed into the state

${\displaystyle |\psi '\rangle =\alpha _{0}|{+}{+}{+}\rangle +\alpha _{1}|{-}{-}{-}\rangle .}$

In the Hadamard basis, bit flips become sign flips and sign flips become bit flips. Let ${\displaystyle E_{\text{phase}}}$ be a quantum channel that can cause at most one phase flip. Then the bit flip code from above can recover ${\displaystyle |\psi \rangle }$ by transforming into the Hadamard basis before and after transmission through ${\displaystyle E_{\text{phase}}}$.

The error channel may induce either a bit flip, a sign flip (i.e., a phase flip), or both. It is possible to correct for both types of errors using one code, and the Shor code does just that. In fact, the Shor code corrects arbitrary single-qubit errors.

Quantum circuit of the Shor code

Let ${\displaystyle E}$ be a quantum channel that can arbitrarily corrupt a single qubit. The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. With the Shor code, a qubit state ${\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }$ will be transformed into the product of 9 qubits ${\displaystyle |\psi '\rangle =\alpha _{0}|0_{S}\rangle +\alpha _{1}|1_{S}\rangle }$, where

${\displaystyle |0_{S}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )}$

${\displaystyle |1_{S}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )}$

If a bit flip error happens to a qubit, the syndrome analysis will be performed on each set of states (1,2,3), (4,5,6), and (7,8,9), then correct the error.

If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. This means that the Shor code can also repair sign flip error for a single qubit.[3]

The Shor code also can correct for any arbitrary errors (both bit flip and sign flip) to a single qubit. If an error is modeled by a unitary transform U, which will act on a qubit ${\displaystyle |\psi \rangle }$, then ${\displaystyle U}$ can be described in the form

${\displaystyle U=c_{0}I+c_{1}\sigma _{x}+c_{2}\sigma _{y}+c_{3}\sigma _{z}}$

where ${\displaystyle c_{0}}$,${\displaystyle c_{1}}$,${\displaystyle c_{2}}$, and ${\displaystyle c_{3}}$ are complex constants, I is the identity, and the Pauli matrices are given by

${\displaystyle \sigma _{x}={\biggl (}{\begin{matrix}0&1\\1&0\end{matrix}}{\biggr )};}$

${\displaystyle \sigma _{y}={\biggl (}{\begin{matrix}0&-i\\i&0\end{matrix}}{\biggr )};}$

${\displaystyle \sigma _{z}={\biggl (}{\begin{matrix}1&0\\0&-1\end{matrix}}{\biggr )}}$

If U is equal to I, then no error occurs. If ${\displaystyle U=\sigma _{x}}$, a bit flip error occurs. If ${\displaystyle U=\sigma _{z}}$, a sign flip error occurs. If ${\displaystyle U=i\sigma _{y}}$ then both a bit flip error and a sign flip error occur. Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors.

Several proposals have been made for storing error-correctable quantum information in bosonic modes. Unlike a two-level system, a quantum harmonic oscillator has infinitely many energy levels in a single physical system. Codes for these systems include cat,[4][5][6] Gottesman-Kitaev-Preskill (GKP),[7] and binomial codes.[8][9] One insight offered by these codes is to take advantage of the redundancy within a single system, rather than to duplicate many two-level qubits.

Written in the Fock basis, the simplest binomial encoding is

${\displaystyle |0_{L}\rangle ={\frac {|0\rangle +|4\rangle }{\sqrt {2}}},\quad |1_{L}\rangle =|2\rangle ,}$

where the subscript L indicates a "logically encoded" state. Then if the dominant error mechanism of the system is the stochastic application of the bosonic lowering operator ${\displaystyle {\hat {a}},}$ the corresponding error states are ${\displaystyle |3\rangle }$ and ${\displaystyle |1\rangle ,}$ respectively. Since the codewords involve only even photon number, and the error states involve only odd photon number, errors can be detected by measuring the photon number parity of the system.[8][10] Measuring the odd parity will allow correction by application of an appropriate unitary operation without knowledge of the specific logical state of the qubit. However, the particular binomial code above is not robust to two-photon loss.

In general, a quantum code for a quantum channel ${\displaystyle {\mathcal {E}}}$ is a subspace ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}}$, where ${\displaystyle {\mathcal {H}}}$ is the state Hilbert space, such that there exists another quantum channel ${\displaystyle {\mathcal {R}}}$ with

${\displaystyle ({\mathcal {R}}\circ {\mathcal {E}})(\rho )=\rho \quad \forall \rho =P_{\mathcal {C}}\rho P_{\mathcal {C}},}$

where ${\displaystyle P_{\mathcal {C}}}$ is the orthogonal projection onto ${\displaystyle {\mathcal {C}}}$. Here ${\displaystyle {\mathcal {R}}}$ is known as the correction operation.

A non-degenerate code is one for which different elements of the set of correctable errors produce linearly independent results when applied to elements of the code. If distinct of the set of correctable errors produce orthogonal results, the code is considered pure.[11]

Over time, researchers have come up with several codes:

• Peter Shor's 9-qubit-code, a.k.a. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a single qubit.
• Andrew Steane found a code which does the same with 7 instead of 9 qubits, see Steane code.
• Raymond Laflamme and collaborators found a class of 5-qubit codes which do the same, which also have the property of being fault-tolerant. A 5-qubit code is the smallest possible code which protects a single logical qubit against single-qubit errors.
• A generalisation of the technique used by Steane, to develop the 7-qubit code from the classical [7, 4] Hamming code, led to the construction of an important class of codes called the CSS codes, named for their inventors: A. R. Calderbank, Peter Shor and Andrew Steane. According to the quantum Hamming bound, encoding a single logical qubit and providing for arbitrary error correction in a single qubit requires a minimum of 5 physical qubits.
• A more general class of codes (encompassing the former) are the stabilizer codes discovered by Daniel Gottesman (), and by A. R. Calderbank, Eric Rains, Peter Shor, and N. J. A. Sloane (, ); these are also called additive codes.
• Two dimensional Bacon-Shor codes are a family of codes parameterized by integers m and n. There are nm qubits arranged in a square lattice.[12]
• A newer idea is Alexei Kitaev's topological quantum codes and the more general idea of a topological quantum computer.
• Todd Brun, Igor Devetak, and Min-Hsiu Hsieh also constructed the entanglement-assisted stabilizer formalism as an extension of the standard stabilizer formalism that incorporates quantum entanglement shared between a sender and a receiver.

That these codes allow indeed for quantum computations of arbitrary length is the content of the quantum threshold theorem, found by Michael Ben-Or and Dorit Aharonov, which asserts that you can correct for all errors if you concatenate quantum codes such as the CSS codes—i.e. re-encode each logical qubit by the same code again, and so on, on logarithmically many levels—provided the error rate of individual quantum gates is below a certain threshold; as otherwise, the attempts to measure the syndrome and correct the errors would introduce more new errors than they correct for.

As of late 2004, estimates for this threshold indicate that it could be as high as 1-3%,[13] provided that there are sufficiently many qubits available.

There have been several experimental realizations of CSS-based codes. The first demonstration was with NMR qubits.[14] Subsequently, demonstrations have been made with linear optics,[15] trapped ions,[16][17] and superconducting (transmon) qubits.[18]

In 2016 for the first time the lifetime of a quantum bit was prolonged by employing a QEC code.[19] The error-correction demonstration was performed on Schrodinger-cat states encoded in a superconducting resonator, and employed a quantum controller capable of performing real-time feedback operations including read-out of the quantum information, its analysis, and the correction of its detected errors. The work demonstrated how the quantum-error-corrected system reaches the break-even point at which the lifetime of a logical qubit exceeds the lifetime of the underlying constituents of the system (the physical qubits).

Other error correcting codes have also been implemented, such as one aimed at correcting for photon loss, the dominant error source in photonic qubit schemes.[20][21]

• Error detection and correction
• Soft error

• Daniel Lidar and Todd Brun, ed. (2013). Quantum Error Correction. Cambridge University Press.
• La Guardia, Giuliano Gadioli, ed. (2020). Quantum Error Correction: Symmetric, Asymmetric, Synchronizable, and Convolutional Codes. Springer Nature.
• Frank Gaitan (2008). Quantum Error Correction and Fault Tolerant Quantum Computing. Taylor & Francis.

• Freedman, Michael H.; Meyer, David A.; Luo, Feng: Z₂-Systolic freedom and quantum codes. Mathematics of quantum computation, 287–320, Comput. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2002.
• Freedman, Michael H.; Meyer, David A. (1998). "Projective plane and planar quantum codes". Found. Comput. Math. 2001 (3): 325–332. arXiv:quant-ph/9810055. Bibcode:1998quant.ph.10055F.
• Lassen, Mikael; Sabuncu, Metin; Huck, Alexander; Niset, Julien; Leuchs, Gerd; Cerf, Nicolas J.; Andersen, Ulrik L. (2010). "Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code". Nature Photonics. 4 (1): 10. Bibcode:2010NaPho...4...10W. doi:10.1038/nphoton.2009.243.

• Knill, E. (2004). "Quantum Computing with Very Noisy Devices". Nature. 434: 39–44. arXiv:quant-ph/0410199. Bibcode:2005Natur.434...39K. doi:10.1038/nature03350. PMID 15744292. S2CID 4420858.
• Error-check breakthrough in quantum computing
• "Topological Quantum Error Correction". Quantum Light. University of Sheffield. September 28, 2018 – via YouTube.