Restrictions on transversal encoded quantum gate sets.

By Bryan Eastin, Emanuel Knill. Physical Review Letters, 2008

531 citations recorded by the source corpus.

Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/59e8fc88297917d9f5ebf385848802c45cb216e7

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Citations in the corpus

Corrigendum: Negative quasi-probability as a resource for quantum computation (2012, background): “Recent theoretical work has shown that a set of quantum gates which is both universal and transversal, and hence fault-tolerant [10, 14, 47], does not exist.”
Fault-tolerant quantum computation with constant overhead (2013, background): “However, transversal gates can never be universal [8].”
Demonstration of fault-tolerant universal quantum gate operations (2021, background): “Combining equations (17)–(19) with the expression for the logical fidelity in equation (20), we can see that expectation values of logical Pauli operators Ot need to be determined to find the logical fidelities”
Many-Body Quantum Magic (2020, background): “Indeed, the Eastin–Knill theorem [4] indicates that no code can implement a universal gate set transversally.”
Negative quasi-probability as a resource for quantum computation (2012, background): “Recent theoretical work has shown that a set of quantum gates which is both universal and transversal, and hence faulttolerant (Zeng et al 2007, Chen et al 2008, Eastin and Knill 2009), does not exist.”
Logical quantum processor based on reconfigurable atom arrays (2023, background): “One important challenge in realizing complex algorithms with logical qubits is that universal computation cannot be implemented trans-versally 42 .”
Single-Shot Fault-Tolerant Quantum Error Correction (2014, background): “Unfortunately, local operations also suffer from limitations, as exemplified by no-go theorems on the computational power of certain classes of local operations [10 – 13].”
Quantum error correction for quantum memories (2013, unknown): “Eastin and Knill (2009) showed that if a quantum code can detect at least any error on a single qubit (meaning that it is a nontrivial code), then it does not have a transversally realizable universal set of gates.”
Quantum error correction: an introductory guide (2019, background): “However, a no-go theorem exists that prohibits the implementation of a full universal gate set in this way on a quantum computer [66].”
Quantum memories at finite temperature (2014, background): “Indeed, quantum coding theory has shown that universal transversal operations are known to be incompatible with stabilizer error correction (Anderson and Jochym-O’Connor, 2014; Eastin and Knill, 2009; Zeng et al., 2011).”
Qudits and High-Dimensional Quantum Computing (2020, methodology): “The π/8 gate, with its close relation to the Clifford group, has many applications and utilities in teleportation-based UQC [66]; transversal implementation [50, 162]; learning an unknown gate [105]; or securing assisted quantum computation [28].”
The resource theory of stabilizer quantum computation (2013, background): “Recent theoretical work has shown that a fault tolerant scheme with a set of quantum gates that is both universal and transversal does not exist[11].”
Noise tailoring for scalable quantum computation via randomized compiling (2015, background): “However, recent ‘nogo’ theorems establish that for any fault-tolerant scheme, there exist some operations that cannot be performed in such a manner [30, 31] and so must be implemented via other means, such as magic-state injection [32] or gauge fixing [33].”
Quantum error mitigation (2022, background): “Unfortunately, it appears that these meth-ods cannot be extended to construct codes of arbitrarily large distances due to the Eastin-Knill theorem (Eastin and Knill, 2009).”
An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation (2009, background): “One might hope to avoid this by ﬁnding a better code, but in fact, no code allows a universal set of transversal gates [ 18 ].”