arXiv:1709.04301

Quantum error correction for non-maximally entangled states

Ritajit Majumdar, Susmita Sur‐Kolay

Published Sep 13, 2017• 0 citations

Abstract

Quantum states have high affinity for errors and hence error correction is of utmost importance to realise a quantum computer. Laflamme showed that 5 qubits are necessary to correct a single error on a qubit. In a Pauli error model, four different types of errors can occur on a qubit. Maximally entangled states are orthogonal to each other and hence can be uniquely distinguished by a measurement in the Bell basis. Thus a measurement in Bell basis and a unitary transformation is sufficient to correct error in Bell states. However, such a measurement is not possible for non-maximally entangled states. In this work we show that the 16 possible errors for a non-maximally entangled two qubit system map to only 8 distinct error states. Hence, it is possible to correct the error without perfect knowledge of the type of error. Furthermore, we show that the possible errors can be grouped in such a way that all 4 errors can occur on one qubit, whereas only bit flip error can occur on the second qubit. As a consequence, instead of 10, only 8 qubits are sufficient to correct a single error. We propose an 8-qubit error correcting code to correct a single error in a non-maximally entangled state. We further argue that for an $n$-qubit non-maximally entangled state of the form $a|0>^{n} + b|1>^{n}$, it is always possible to correct a single error with fewer than $5n$ qubits, in fact only $3n+2$ qubits suffice.

DOI: 10.48550/arxiv.1709.04301

Related papers

Quantum error-correcting code for ternary logic

Ritajit Majumdar, Saikat Basu, Shibashis Ghosh +1 more

arXiv:1807.01863Physical Review AMay 2, 201826 citations

Quantum Error Correction

WORLD SCIENTIFIC eBooksFeb 20, 20183 citations

Quantum error correction

WORLD SCIENTIFIC eBooksDec 1, 20180 citations

Scaling Up Bit-Flip Quantum Error Correction

Farhan Feroz, A. B. M. Alim Al Islam

International Conference on Networking, Systems and Security (NSysS)Dec 22, 20201 citation

Experimental verification of five-qubit quantum error correction with superconducting qubits

Ming Gong, Xiao Yuan, Shiyu Wang +19 more

Jul 10, 201916 citations

Beyond Single-Shot Fault-Tolerant Quantum Error Correction

Nicolas Delfosse, Ben W. Reichardt, Krysta M. Svore

arXiv:2002.05180IEEE Transactions on Information TheoryOct 16, 202130 citations

arXiv:1709.04301

Quantum error correction for non-maximally entangled states

Ritajit Majumdar, Susmita Sur‐Kolay

Published Sep 13, 2017• 0 citations

arXiv HTML PDF

Abstract

n

-qubit non-maximally entangled state of the form

a|0>^{n} + b|1>^{n}

, it is always possible to correct a single error with fewer than

5n

qubits, in fact only

3n+2

qubits suffice.

DOI: 10.48550/arxiv.1709.04301

Related papers

Quantum error-correcting code for ternary logic

Ritajit Majumdar, Saikat Basu, Shibashis Ghosh +1 more

Ternary quantum systems are being studied because they provide more computational state space per unit of information, known as qutrit. A qutrit has three basis states, thus a qubit may be considered as a special case of a qutrit where the coefficient of one of the basis states is zero. Hence both $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional and $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional as well as $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors in qutrits and show that any pairwise bit swap error can be expressed as a linear combination of shift errors and phase errors, (ii) propose a special type of error called a quantum superposition error and show its equivalence to arbitrary rotation, (iii) formulate a nine-qutrit code which can correct a single error in a qutrit, and (iv) provide its stabilizer and circuit realization.Ternary quantum systems are being studied because they provide more computational state space per unit of information, known as qutrit. A qutrit has three basis states, thus a qubit may be considered as a special case of a qutrit where the coefficient of one of the basis states is zero. Hence both $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional and $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional as well as $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors in qutrits and show that any pairwise bit swap error can be expressed as a linear combination of shift errors and phase errors, (ii) propose a special type of error called a quantum superposition error and show its equivalence to arbitrary rotation, (iii) formulate a nine-qutrit code which can correct a single error in a qutrit, and (iv) provide its stabilizer and circuit realization.

arXiv:1807.01863Physical Review AMay 2, 201826 citations

Quantum Error Correction

Quantum error correction is a set of methods to protect quantum information--that is, quantum states--from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space, without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design.Quantum error correction is a set of methods to protect quantum information--that is, quantum states--from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space, without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design.

WORLD SCIENTIFIC eBooksFeb 20, 20183 citations

Quantum error correction

WORLD SCIENTIFIC eBooksDec 1, 20180 citations

Scaling Up Bit-Flip Quantum Error Correction

Farhan Feroz, A. B. M. Alim Al Islam

Quantum error correction comes from the marriage of quantum mechanics with the theory of error-correcting codes, which is still at the rudimentary stage tackling only 1 qubit to date. As, unlike the classical world, the extent of error in the quantum world is substantially high and unavoidable, scaling up the quantum error correction from 1 qubit to higher is of utmost significane. Therefore, to achive the scaling up, we present a general quantum error correction code that can be adapted for any number of quantum errors (bit-flip error to be specific in our case). This generalized code for bit-flip error correction realizes a delicate blending of using controlled-NOT gates with the notion of quantum entanglement. We augment the blending with majority voting in the final stage of the error correction. We demonstrate efficacy of our proposed error correction code through real experimentation in IBM Qiskit. Our experimental results up to 3-qubit bit-flip errors in the available Qiskit infrastructure confirm a total escape from errors using our proposed error correction code.Quantum error correction comes from the marriage of quantum mechanics with the theory of error-correcting codes, which is still at the rudimentary stage tackling only 1 qubit to date. As, unlike the classical world, the extent of error in the quantum world is substantially high and unavoidable, scaling up the quantum error correction from 1 qubit to higher is of utmost significane. Therefore, to achive the scaling up, we present a general quantum error correction code that can be adapted for any number of quantum errors (bit-flip error to be specific in our case). This generalized code for bit-flip error correction realizes a delicate blending of using controlled-NOT gates with the notion of quantum entanglement. We augment the blending with majority voting in the final stage of the error correction. We demonstrate efficacy of our proposed error correction code through real experimentation in IBM Qiskit. Our experimental results up to 3-qubit bit-flip errors in the available Qiskit infrastructure confirm a total escape from errors using our proposed error correction code.

International Conference on Networking, Systems and Security (NSysS)Dec 22, 20201 citation

Experimental verification of five-qubit quantum error correction with superconducting qubits

Ming Gong, Xiao Yuan, Shiyu Wang +19 more

Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correction code (QECC), with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this notoriously difficult challenge, we push the limits of the depth of superconducting quantum circuits and experimentally realise the universal five-qubit QECC, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the five-qubit QECC for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\%$ while with a high fidelity of $98.6(1)\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of $97.2(2)\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with a process fidelity of $57.4(7)\%$ . After decoding, by identifying errors via the measurement of the four ancillae and then mathematically recovering the qubit state, we verify the power of error correction of this code. Thus, by demonstrating each key aspect of error correction with the five-qubit code, our work establishes the viability of experimental quantum error correction with superconducting qubits and paves the route to fault-tolerant quantum computing.Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correction code (QECC), with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this notoriously difficult challenge, we push the limits of the depth of superconducting quantum circuits and experimentally realise the universal five-qubit QECC, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the five-qubit QECC for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\%$ while with a high fidelity of $98.6(1)\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of $97.2(2)\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with a process fidelity of $57.4(7)\%$. After decoding, by identifying errors via the measurement of the four ancillae and then mathematically recovering the qubit state, we verify the power of error correction of this code. Thus, by demonstrating each key aspect of error correction with the five-qubit code, our work establishes the viability of experimental quantum error correction with superconducting qubits and paves the route to fault-tolerant quantum computing.

Jul 10, 201916 citations

Beyond Single-Shot Fault-Tolerant Quantum Error Correction

Nicolas Delfosse, Ben W. Reichardt, Krysta M. Svore

Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requires $\Omega (r d^{2})$ parity check measurements for a distance- $d$ code defined by $r$ parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only $r$ measurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rate $p$ . For this model, we demonstrate that fault-tolerant quantum error correction can be achieved using $O(d \log (d))$ measurements for any code with distance $d \geq \Omega (n^\alpha)$ for some constant $\alpha > 0$ . Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than $r$ measurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requires$\Omega (r d^{2})$parity check measurements for a distance-$d$code defined by$r$parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only$r$measurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rate$p$. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved using$O(d \log (d))$measurements for any code with distance$d \geq \Omega (n^\alpha)$for some constant$\alpha > 0$. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than$r$measurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.

arXiv:2002.05180IEEE Transactions on Information TheoryOct 16, 202130 citations

n

-qubit non-maximally entangled state of the form

a|0>^{n} + b|1>^{n}

, it is always possible to correct a single error with fewer than

5n

qubits, in fact only

3n+2

qubits suffice.

57.1(3)\%

while with a high fidelity of

98.6(1)\%

in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of

97.2(2)\%

within the code space. Finally, we realise the decoding circuit and recover the input state with a process fidelity of

57.4(7)\%

. After decoding, by identifying errors via the measurement of the four ancillae and then mathematically recovering the qubit state, we verify the power of error correction of this code. Thus, by demonstrating each key aspect of error correction with the five-qubit code, our work establishes the viability of experimental quantum error correction with superconducting qubits and paves the route to fault-tolerant quantum computing.

\Omega (r d^{2})

parity check measurements for a distance-

d

code defined by

r

parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only

r

measurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rate

p

. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved using

O(d \log (d))

measurements for any code with distance

d \geq \Omega (n^\alpha)

for some constant

\alpha > 0

. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than

r

measurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.