More than a half-century after the classical discovery of an equivalence between insertion and deletion errors, researchers at the University of Sheffield have made progress towards a quantum counterpart, addressing a longstanding problem in quantum error correction. Lewis Bulled and Yingkai Ouyang detail the conditions under which permutation-invariant codes can correct errors that simultaneously delete and insert qubits, a more complex scenario than addressing each error type separately.
Their work resolves many outstanding questions regarding the quantum error correction of synchronisation errors, which change the number of qubits in a quantum system. The researchers formulate conditions under which these codes are (t,s)-insdel error-correctable.
Quantum Insertion-Deletion Equivalence on Permutation-Invariant Codes
Permutation-invariant codes can now correct errors arising from both the deletion and insertion of qubits, a result demonstrated by Lewis Bulled and Yingkai Ouyang at the University of Sheffield. Their work establishes a direct equivalence between a code’s ability to withstand deletion errors and its capacity to correct insertion errors, a connection previously understood in classical error correction more than a half-century ago but elusive in the quantum realm. Nakayama and Hagiwara presented the first quantum deletion code, an eight-qubit code capable of correcting a single deletion error, and subsequently reduced the code length to four qubits.
The authors argued that the optimal code length is four qubits, that is, there exists no quantum code on two or three qubits capable of correcting a single deletion error, and later provided a general construction of such codes.
Bulled and Ouyang extended this foundation by proving the quantum insertion-deletion equivalence on permutation-invariant codes, using combinatorial techniques to show that conditions for t-insertion error correction are directly linked to *2t*-deletion conditions established by Aydin et al. This equivalence means a code capable of correcting up to t deletions can also correct up to t insertions, and vice versa.
The researchers detail these conditions in Theorem 1, which establishes that a permutation-invariant code with logical coefficients α and β is t-insertion error-correctable if and only if certain criteria, denoted as (C1) and (C2), are met for all values of a’, b’, and j within specified ranges.
Synchronisation Errors in Quantum Communication and QEC
Unlike many areas of quantum error correction focused on bit-flip or phase-flip errors, the study highlights a historical under-emphasis on synchronisation errors despite their prevalence in quantum communication systems, particularly those employing optical channels where qubit loss is common. This work directly addresses a gap in the field, building on classical understandings of insertion-deletion equivalence that originated over a half-century ago but lacked a quantum counterpart until now.
The team’s analysis extends beyond simple qubit deletion to encompass quantum insertion errors, where unwanted qubits are added to the logical state, represented mathematically as a sum over all possible insertion configurations. These insertion errors are described by which define the probability amplitude of each possible qubit insertion. The Sheffield researchers formulated conditions for correcting these errors within the framework of permutation-invariant codes.
This equivalence is particularly relevant because of how permutation-invariant codes are defined within the symmetric space, a subspace of a Hilbert space where states remain unchanged under qubit permutations. These codes use Dicke states, which are specific quantum states defined by the number of qubits in a given configuration, and are expressed as a convex combination of these states.
Shibayama and Hagiwara introduced a method for constructing permutation-invariant (PI) codes capable of correcting multiple deletions. Ouyang noted an equivalence between erasure and deletion errors on PI codes, and proposed that such codes must have distance at least t + 1 to correct t deletions. Aydin et al. proposed an infinite family of PI codes, so-called combinatorial or AAB codes, which can correct 2t deletion errors (and thus all t-qubit Pauli errors.
Nakayama-Hagiwara Quantum Deletion Codes and PI Code Construction
Establishing a direct link between correcting qubit deletions and insertions within quantum systems represents an advance in error mitigation, as demonstrated by work from the University of Sheffield. Researchers proved the quantum insertion-deletion equivalence for permutation-invariant (PI) codes by establishing a set of t-insertion conditions, and subsequently showing these are equivalent to the 2*t*-deletion conditions detailed by Aydin et al.
Theorem 1, central to the study, formally defines the conditions under which PI codes are t-insertion error-correctable, linking these conditions directly to established deletion error correction criteria. The team extended these conditions to account for quantum insdel errors, formulating a more restrictive set of (t,s)-insdel conditions, acknowledging that correcting both insertions and deletions simultaneously presents a greater challenge. This refinement acknowledges that the requirements for correcting both error types are more demanding than those for addressing either deletion or insertion in isolation.
Dicke States and Encoding in Permutation-Invariant Codes
Permutation-invariant codes use Dicke states, symmetric quantum states, as a foundational element for encoding information, which simplifies error correction within the system. These codes, defined within a Hilbert space remaining unchanged under qubit permutations, use Dicke states to construct logical states as convex combinations, where k represents a weighting determined by coefficients c0 and c1.
This construction allows for a robust encoding scheme, particularly relevant when addressing the challenges posed by quantum synchronisation errors. This advancement addresses a longstanding gap in quantum error correction, as classical counterparts have understood insertion-deletion equivalence for over a half-century, but a quantum equivalent remained elusive.
The researchers built upon earlier methods for constructing PI codes capable of correcting multiple deletions, noting that a code must possess a distance of at least t + 1 to correct t deletions. The sum of the squared magnitudes of these coefficients must equal one, ensuring the state remains normalized.
This formulation allows for a subtle analysis of how insertions impact the encoded information. The team demonstrates that the composition of insertion and deletion channels can be understood through the number of inserted qubits subsequently removed, enabling a commutation between the resulting channels. This decomposition is important for designing codes capable of handling both types of errors simultaneously, a scenario more complex than addressing them in isolation.
Quantum Insertion Channels and Operator Formalism
Quantum insertion errors, where extraneous qubits unexpectedly appear within a quantum state, are mathematically described using insertion coefficients, a framework detailed in recent work from the University of Sheffield. This decomposition is important for designing codes capable of handling both types of errors simultaneously, a scenario more complex than addressing them in isolation.
Specifically, the work introduces a channel defined as acting on a quantum state *σ*, revealing a positive semidefinite operator important for establishing error correction protocols. This is achieved by defining a probability distribution r and w and demonstrating that the resulting operator, defining a quantum channel, maintains trace preservation.
The researchers show that for all PI states, these distributions exist, ensuring the viability of the correction process. The paper states, “Thm. 1 states the conditions on α,β for which the PI code given by (2) is t-insertion error-correctable.” The team’s approach not only clarifies the mathematical underpinnings of synchronisation errors but also provides a pathway toward constructing more resilient quantum communication systems.
The resulting operator, the researchers note, must have a trace lying within the closed unit interval, a condition essential for ensuring the channel’s validity and the overall stability of the quantum state. This detailed analysis of error composition and correction conditions represents an advancement in the field.
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