scholarly journals EFFICIENT QUANTUM CIRCUITS FOR NON-QUBIT QUANTUM ERROR-CORRECTING CODES

2003 ◽  
Vol 14 (05) ◽  
pp. 757-775 ◽  
Author(s):  
MARKUS GRASSL ◽  
MARTIN RÖTTELER ◽  
THOMAS BETH
Keyword(s):  
Prime Power ◽  
Tensor Products ◽  
Quantum Circuit ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Quantum Circuits ◽  
Running Time ◽  
Classical Algorithm ◽  
Quantum Error

We present two methods for the construction of quantum circuits for quantum error- correcting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n - k)) gates. The running time of the classical algorithm to compute the quantum circuit is O(n(n - k)2).

scholarly journals ON OPTIMAL QUANTUM CODES

2004 ◽  
Vol 02 (01) ◽  
pp. 55-64 ◽  
Author(s):  
MARKUS GRASSL ◽  
THOMAS BETH ◽  
MARTIN RÖTTELER
Keyword(s):  
Minimum Distance ◽  
Prime Power ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Codes ◽  
Maximum Distance Separable ◽  
Shortened Codes ◽  
Quantum Error ◽  
Quantum Mds Codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.

scholarly journals Online scheduled execution of quantum circuits protected by surface codes

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1335-1348 ◽  
Author(s):  
Alexandru Paler ◽  
Austin G. Fowler ◽  
Robert Wille
Keyword(s):  
Quantum Information Processing ◽  
Fault Tolerant ◽  
Dynamic Structure ◽  
Quantum Circuit ◽  
Error Correcting Codes ◽  
Quantum Circuits ◽  
Static Scheduling ◽  
Surface Code ◽  
High Level ◽  
Quantum Error

Quantum circuits are the preferred formalism for expressing quantum information processing tasks. Quantum circuit design automation methods mostly use a waterfall approach and consider that high level circuit descriptions are hardware agnostic. This assumption has lead to a static circuit perspective: the number of quantum bits and quantum gates is determined before circuit execution and everything is considered reliable with zero probability of failure. Many different schemes for achieving reliable fault-tolerant quantum computation exist, with different schemes suitable for different architectures. A number of large experimental groups are developing architectures well suited to being protected by surface quantum error correcting codes. Such circuits could include unreliable logical elements, such as state distillation, whose failure can be determined only after their actual execution. Therefore, practical logical circuits, as envisaged by many groups, are likely to have a dynamic structure. This requires an online scheduling of their execution: one knows for sure what needs to be executed only after previous elements have finished executing. This work shows that scheduling shares similarities with place and route methods. The work also introduces the first online schedulers of quantum circuits protected by surface codes. The work also highlights scheduling efficiency by comparing the new methods with state of the art static scheduling of surface code protected fault-tolerant circuits.

scholarly journals Universal Framework for Quantum Error-Correcting Codes

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 937
Author(s):  
Zhuo Li ◽  
Lijuan Xing
Keyword(s):  
Group Algebra ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Algebraic Notation ◽  
General Quantum ◽  
Quantum Error

We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.

scholarly journals Fault-Tolerant quantum computation with constant overhead

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1339-1371
Author(s):  
Daniel Gottesman
Keyword(s):  
Fault Tolerant ◽  
Quantum Circuit ◽  
Error Correcting Codes ◽  
Parity Check ◽  
Low Density Parity Check ◽  
The Family ◽  
Minimum Number ◽  
Logical Qubits ◽  
Quantum Error ◽  
Parity Check Codes

What is the minimum number of extra qubits needed to perform a large fault-tolerant quantum circuit? Working in a common model of fault-tolerance, I show that in the asymptotic limit of large circuits, the ratio of physical qubits to logical qubits can be a constant. The construction makes use of quantum low-density parity check codes, and the asymptotic overhead of the protocol is equal to that of the family of quantum error-correcting codes underlying the fault-tolerant protocol.

New nonbinary quantum codes with larger distance constructed from BCH codes over 𝔽q2

2017 ◽  
Vol 31 (06) ◽  
pp. 1750034 ◽  
Author(s):  
Gen Xu ◽  
Ruihu Li ◽  
Qiang Fu ◽  
Yuena Ma ◽  
Luobin Guo
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Careful Analysis ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Quantum Codes ◽  
Code Length ◽  
Narrow Sense ◽  
Cyclotomic Cosets ◽  
Quantum Error

This paper concentrates on construction of new nonbinary quantum error-correcting codes (QECCs) from three classes of narrow-sense imprimitive BCH codes over finite field [Formula: see text] ([Formula: see text] is an odd prime power). By a careful analysis on properties of cyclotomic cosets in defining set [Formula: see text] of these BCH codes, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing BCH codes is determined to be much larger than the result given according to Aly et al. [S. A. Aly, A. Klappenecker and P. K. Sarvepalli, IEEE Trans. Inf. Theory 53, 1183 (2007)] for each different code length. Thus families of new nonbinary QECCs are constructed, and the newly obtained QECCs have larger distance than those in previous literature.

scholarly journals Estimating Gibbs partition function with quantum Clifford sampling

2022 ◽  
Author(s):  
Yusen Wu ◽  
Jingbo B Wang
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Sampling Technique ◽  
Quantum Circuit ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Partition Functions ◽  
Stochastic Matrices ◽  
Many Body ◽  
Quantum Devices

Abstract The partition function is an essential quantity in statistical mechanics, and its accurate computation is a key component of any statistical analysis of quantum system and phenomenon. However, for interacting many-body quantum systems, its calculation generally involves summing over an exponential number of terms and can thus quickly grow to be intractable. Accurately and efficiently estimating the partition function of its corresponding system Hamiltonian then becomes the key in solving quantum many-body problems. In this paper we develop a hybrid quantumclassical algorithm to estimate the partition function, utilising a novel Clifford sampling technique. Note that previous works on quantum estimation of partition functions require O(1/ε√∆)-depth quantum circuits [17, 23], where ∆ is the minimum spectral gap of stochastic matrices and ε is the multiplicative error. Our algorithm requires only a shallow O(1)-depth quantum circuit, repeated O(n/ε2) times, to provide a comparable ε approximation. Shallow-depth quantum circuits are considered vitally important for currently available NISQ (Noisy Intermediate-Scale Quantum) devices.

scholarly journals Hybrid Quantum-Classical Eigensolver without Variation or Parametric Gates

2021 ◽  
Vol 3 (1) ◽  
pp. 137-152
Author(s):  
Pejman Jouzdani ◽  
Stefan Bringuier
Keyword(s):  
Excited States ◽  
Numerical Algorithms ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Effective Hamiltonian ◽  
Quantum Devices ◽  
Near Term ◽  
Classical Computer ◽  
Quantum Error ◽  
H2 Molecule

The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here, we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection, an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground state and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H2 molecule.

New optimal asymmetric quantum codes constructed from constacyclic codes

2017 ◽  
Vol 31 (05) ◽  
pp. 1750030 ◽  
Author(s):  
Gen Xu ◽  
Ruihu Li ◽  
Luobin Guo ◽  
Liangdong Lü
Keyword(s):  
Prime Power ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Code Length ◽  
Theor Phys ◽  
Singleton Bound ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes ◽  
Quantum Error

In this paper, we propose the construction of asymmetric quantum codes from two families of constacyclic codes over finite field [Formula: see text] of code length [Formula: see text], where for the first family, [Formula: see text] is an odd prime power with the form [Formula: see text] ([Formula: see text] is integer) or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]; for the second family, [Formula: see text] is an odd prime power with the form [Formula: see text] or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]. As a result, families of new asymmetric quantum codes [Formula: see text] with [Formula: see text] distance larger than [Formula: see text] are obtained, which are not covered by the asymmetric quantum error-correcting codes (AQECCs) in Refs. 32 and 33 [J.-Z. Chen, J.-P. Li and J. Lin, Int. J. Theor. Phys. 53, 72 (2014); L. Wang and S. Zhu, Int. J. Quantum Inf. 12, 1450017 (2014)] that [Formula: see text]. Also, all the newly obtained asymmetric quantum codes are optimal according to the singleton bound for asymmetric quantum codes.

scholarly journals Connectivity matrix model of quantum circuits and its application to distributed quantum circuit optimization

2021 ◽  
Vol 20 (7) ◽  
Author(s):  
Ismail Ghodsollahee ◽  
Zohreh Davarzani ◽  
Mariam Zomorodi ◽  
Paweł Pławiak ◽  
Monireh Houshmand ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Connectivity Matrix ◽  
Circuit Optimization ◽  
Novel Approach ◽  
The Matrix ◽  
Minimum Number ◽  
Two Phases

AbstractAs quantum computation grows, the number of qubits involved in a given quantum computer increases. But due to the physical limitations in the number of qubits of a single quantum device, the computation should be performed in a distributed system. In this paper, a new model of quantum computation based on the matrix representation of quantum circuits is proposed. Then, using this model, we propose a novel approach for reducing the number of teleportations in a distributed quantum circuit. The proposed method consists of two phases: the pre-processing phase and the optimization phase. In the pre-processing phase, it considers the bi-partitioning of quantum circuits by Non-Dominated Sorting Genetic Algorithm (NSGA-III) to minimize the number of global gates and to distribute the quantum circuit into two balanced parts with equal number of qubits and minimum number of global gates. In the optimization phase, two heuristics named Heuristic I and Heuristic II are proposed to optimize the number of teleportations according to the partitioning obtained from the pre-processing phase. Finally, the proposed approach is evaluated on many benchmark quantum circuits. The results of these evaluations show an average of 22.16% improvement in the teleportation cost of the proposed approach compared to the existing works in the literature.

[[2n,n]] Additive quantum error correcting codes

2008 ◽  
Vol 25 (4) ◽  
pp. 519-522
Author(s):  
Yongjun Du ◽  
Yuefei Ma
Keyword(s):  
Error Correcting Codes ◽  
Quantum Error
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