Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0405038

Creating Boolean Functions for the Five-EPR-Pair, Single-Error-Correcting Code

Jin-Yuan Hsieh, Che‐Ming Li

Published May 8, 2004• 0 citations
arXivHTMLPDF

Abstract

A quantum single-error-correcting scheme can be derived from a one-way entanglement purification protocol in purifying one Bell state from a finite block of five Bell states. The main issue to be concerned with in the theory of such an error-correction is to create specific linear Boolean functions that can transform the sixteen error syndromes occurring in the error-correcting code onto their mappings so that one Bell state is corrected whenever the other four in the finite block are measured. The Boolean function is performed under the effect of its associated sequence of basic quantum unilateral and bilateral operations. Previously, the Boolean function is created in use of the Monte Carlo computer search method. We introduce here a systematic scenario for creating the Boolean function and its associated sequence of operations so that we can do the job in an analytical way without any trial and error effort. Consequently, all possible Boolean functions can in principle be created by using our method. Furthermore, for a deduced Boolean function, we can also in the spirit of our method derive its best associated sequence of operations which may contain the least number of total operations or the least number of the bilateral XOR operations alone. Some results obtained in this work show the capability of our method in creating the Boolean function and its sequence of operations.

DOI
10.48550/arxiv.quant-ph/0405038

Related papers

Analytical technique for simplification of the encoder–decoder circuit for a perfect five-qubit error correction

Jin-Yuan Hsieh, Che-Ming Li, Der-San Chuu

Simpler encoding and decoding networks are necessary for more reliable quantum error-correcting codes (QECCs). The simplification of the encoder–decoder circuit for a perfect five-qubit QECC can be derived analytically if the QECC is converted from its equivalent one-way entanglement purification protocol. In this work, the analytical method to simplify the encoder–decoder circuit is introduced and a circuit that is as simple as the existing simplest circuits is presented as an example. The encoder–decoder circuit presented here involves nine single- and two-qubit unitary operations, only six of which are controlled-NOT gates.

arXiv:quant-ph/0410004•New Journal of Physics•May 30, 2006•2 citations

Five-wave-packet linear optics quantum-error-correcting code

Thomas Alfred Walker, Samuel L. Braunstein

In this article we outline a method for generating linear optics circuits that encode quantum-error-correcting codes. Using this method we produce a single-error-correcting code encoding one wave packet over five which can be implemented using linear optics and feed-forward correction. This code improves on the capacity of the best known code that can be implemented using linear optics and saturates the lower bound for the number of carriers needed for a single-error-correcting code. Our code can correct arbitrary single errors that occur randomly on each wave packet corresponding to a non-Gaussian error model, thus circumventing the so-called no-go theorem for Gaussian quantum-error correction.

Physical Review A•Jun 4, 2010•8 citations

Short Shor-style syndrome sequences

Nicolas Delfosse, Ben W. Reichardt

We optimize fault-tolerant quantum error correction to reduce the number of syndrome bit measurements. Speeding up error correction will also speed up an encoded quantum computation, and should reduce its effective error rate. We give both code-specific and general methods, using a variety of techniques and in a variety of settings. We design new quantum error-correcting codes specifically for efficient error correction, e.g., allowing single-shot error correction. For codes with multiple logical qubits, we give methods for combining error correction with partial logical measurements. There are tradeoffs in choosing a code and error-correction technique. While to date most work has concentrated on optimizing the syndrome-extraction procedure, we show that there are also substantial benefits to optimizing how the measured syndromes are chosen and used. As an example, we design single-shot measurement sequences for fault-tolerant quantum error correction with the 16-qubit extended Hamming code. Our scheme uses 10 syndrome bit measurements, compared to 40 measurements with the Shor scheme. We design single-shot logical measurements as well: any logical Z measurement can be made together with fault-tolerant error correction using only 11 measurements. For comparison, using the Shor scheme a basic implementation of such a non-destructive logical measurement uses 63 measurements. We also offer ten open problems, the solutions of which could lead to substantial improvements of fault-tolerant error correction.

arXiv:2008.05051•Aug 12, 2020•4 citations

Interoperability in encoded quantum repeater networks

Shota Nagayama, Byung-Soo Choi, Simon J. Devitt +2 more

The future of quantum repeater networking will require interoperability between various error-correcting codes. A few specific code conversions and even a generalized method are known, however, no detailed analysis of these techniques in the context of quantum networking has been performed. In this paper we analyze a generalized procedure to create Bell pairs encoded heterogeneously between two separate codes used often in error-corrected quantum repeater network designs. We begin with a physical Bell pair and then encode each qubit in a different error-correcting code, using entanglement purification to increase the fidelity. We investigate three separate protocols for preparing the purified encoded Bell pair. We calculate the error probability of those schemes between the Steane [[7,1,3]] code, a distance-3 surface code, and single physical qubits by Monte Carlo simulation under a standard Pauli error model and estimate the resource efficiency of the procedures. A local gate error rate of ${10}^{\ensuremath{-}3}$ allows us to create high-fidelity logical Bell pairs between any of our chosen codes. We find that a postselected model, where any detected parity flips in code stabilizers result in a restart of the protocol, performs the best.

arXiv:1508.04599•Physical Review A•Apr 26, 2016•14 citations

Fault-Tolerant Error Correction with Efficient Quantum Codes

David Peter Divincenzo, Peter W. Shor

We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced group-theoretic framework for unifying all known quantum codes.

arXiv:quant-ph/9605031•Physical Review Letters•Oct 7, 1996•337 citations

Sequential measurements, disturbance and property testing

Aram W. Harrow, Cedric Yen-Yu Lin, Ashley Montanaro

We describe two procedures which, given access to one copy of a quantum state and a sequence of two-outcome measurements, can distinguish between the case that at least one of the measurements accepts the state with high probability, and the case that all of the measurements have low probability of acceptance. The measurements cannot simply be tried in sequence, because early measurements may disturb the state being tested. One procedure is based on a variant of Marriott-Watrous amplification. The other procedure is based on the use of a test for this disturbance, which is applied with low probability. We find a number of applications: Quantum query complexity separations in the property testing model for testing isomorphism of functions under group actions. We give quantum algorithms for testing isomorphism, linear isomorphism and affine isomorphism of boolean functions which use exponentially fewer queries than is possible classically, and a quantum algorithm for testing graph isomorphism which uses polynomially fewer queries than the best algorithm known. Testing properties of quantum states and operations. We show that any finite property of quantum states can be tested using a number of copies of the state which is logarithmic in the size of the property, and give a test for genuine multipartite entanglement of states of n qubits that uses O(n) copies of the state. We also show that equivalence of two unitary operations under conjugation by a unitary picked from a fixed set can be tested efficiently. This is a natural quantum generalisation of testing isomorphism of boolean functions. Correcting an error in a result of Aaronson on de- Merlinizing quantum protocols. This result claimed that, in any one-way quantum communication protocol where two parties are assisted by an all-powerful but untrusted third party, the third party can be removed with only a modest increase in the communication cost. We give a corrected proof of a key technical lemma required for Aaronson's result.

arXiv:1607.03236•ACM-SIAM Symposium on Discrete Algorithms (SODA)•Jan 1, 2017•2 citations
A quantum single-error-correcting scheme can be derived from a one-way entanglement purification protocol in purifying one Bell state from a finite block of five Bell states. The main issue to be concerned with in the theory of such an error-correction is to create specific linear Boolean functions that can transform the sixteen error syndromes occurring in the error-correcting code onto their mappings so that one Bell state is corrected whenever the other four in the finite block are measured. The Boolean function is performed under the effect of its associated sequence of basic quantum unilateral and bilateral operations. Previously, the Boolean function is created in use of the Monte Carlo computer search method. We introduce here a systematic scenario for creating the Boolean function and its associated sequence of operations so that we can do the job in an analytical way without any trial and error effort. Consequently, all possible Boolean functions can in principle be created by using our method. Furthermore, for a deduced Boolean function, we can also in the spirit of our method derive its best associated sequence of operations which may contain the least number of total operations or the least number of the bilateral XOR operations alone. Some results obtained in this work show the capability of our method in creating the Boolean function and its sequence of operations.
Simpler encoding and decoding networks are necessary for more reliable quantum error-correcting codes (QECCs). The simplification of the encoder–decoder circuit for a perfect five-qubit QECC can be derived analytically if the QECC is converted from its equivalent one-way entanglement purification protocol. In this work, the analytical method to simplify the encoder–decoder circuit is introduced and a circuit that is as simple as the existing simplest circuits is presented as an example. The encoder–decoder circuit presented here involves nine single- and two-qubit unitary operations, only six of which are controlled-NOT gates.
In this article we outline a method for generating linear optics circuits that encode quantum-error-correcting codes. Using this method we produce a single-error-correcting code encoding one wave packet over five which can be implemented using linear optics and feed-forward correction. This code improves on the capacity of the best known code that can be implemented using linear optics and saturates the lower bound for the number of carriers needed for a single-error-correcting code. Our code can correct arbitrary single errors that occur randomly on each wave packet corresponding to a non-Gaussian error model, thus circumventing the so-called no-go theorem for Gaussian quantum-error correction.
We optimize fault-tolerant quantum error correction to reduce the number of syndrome bit measurements. Speeding up error correction will also speed up an encoded quantum computation, and should reduce its effective error rate. We give both code-specific and general methods, using a variety of techniques and in a variety of settings. We design new quantum error-correcting codes specifically for efficient error correction, e.g., allowing single-shot error correction. For codes with multiple logical qubits, we give methods for combining error correction with partial logical measurements. There are tradeoffs in choosing a code and error-correction technique. While to date most work has concentrated on optimizing the syndrome-extraction procedure, we show that there are also substantial benefits to optimizing how the measured syndromes are chosen and used. As an example, we design single-shot measurement sequences for fault-tolerant quantum error correction with the 16-qubit extended Hamming code. Our scheme uses 10 syndrome bit measurements, compared to 40 measurements with the Shor scheme. We design single-shot logical measurements as well: any logical Z measurement can be made together with fault-tolerant error correction using only 11 measurements. For comparison, using the Shor scheme a basic implementation of such a non-destructive logical measurement uses 63 measurements. We also offer ten open problems, the solutions of which could lead to substantial improvements of fault-tolerant error correction.
The future of quantum repeater networking will require interoperability between various error-correcting codes. A few specific code conversions and even a generalized method are known, however, no detailed analysis of these techniques in the context of quantum networking has been performed. In this paper we analyze a generalized procedure to create Bell pairs encoded heterogeneously between two separate codes used often in error-corrected quantum repeater network designs. We begin with a physical Bell pair and then encode each qubit in a different error-correcting code, using entanglement purification to increase the fidelity. We investigate three separate protocols for preparing the purified encoded Bell pair. We calculate the error probability of those schemes between the Steane [[7,1,3]] code, a distance-3 surface code, and single physical qubits by Monte Carlo simulation under a standard Pauli error model and estimate the resource efficiency of the procedures. A local gate error rate of ${10}^{\ensuremath{-}3}$ allows us to create high-fidelity logical Bell pairs between any of our chosen codes. We find that a postselected model, where any detected parity flips in code stabilizers result in a restart of the protocol, performs the best.
We describe two procedures which, given access to one copy of a quantum state and a sequence of two-outcome measurements, can distinguish between the case that at least one of the measurements accepts the state with high probability, and the case that all of the measurements have low probability of acceptance. The measurements cannot simply be tried in sequence, because early measurements may disturb the state being tested. One procedure is based on a variant of Marriott-Watrous amplification. The other procedure is based on the use of a test for this disturbance, which is applied with low probability. We find a number of applications: Quantum query complexity separations in the property testing model for testing isomorphism of functions under group actions. We give quantum algorithms for testing isomorphism, linear isomorphism and affine isomorphism of boolean functions which use exponentially fewer queries than is possible classically, and a quantum algorithm for testing graph isomorphism which uses polynomially fewer queries than the best algorithm known. Testing properties of quantum states and operations. We show that any finite property of quantum states can be tested using a number of copies of the state which is logarithmic in the size of the property, and give a test for genuine multipartite entanglement of states of n qubits that uses O(n) copies of the state. We also show that equivalence of two unitary operations under conjugation by a unitary picked from a fixed set can be tested efficiently. This is a natural quantum generalisation of testing isomorphism of boolean functions. Correcting an error in a result of Aaronson on de- Merlinizing quantum protocols. This result claimed that, in any one-way quantum communication protocol where two parties are assisted by an all-powerful but untrusted third party, the third party can be removed with only a modest increase in the communication cost. We give a corrected proof of a key technical lemma required for Aaronson's result.