scholarly journals Accuracy threshold for postselected quantum computation

2008 ◽  
Vol 8 (3&4) ◽  
pp. 181-244 ◽  
Author(s):  
P. Aliferis ◽  
D. Gottesman ◽  
J. Preskill
Keyword(s):  
Lower Bound ◽  
Quantum Computation ◽  
Error Detection ◽  
Fault Tolerant ◽  
Error Correcting Codes ◽  
Stochastic Noise ◽  
Strongly Correlated ◽  
Error Detecting ◽  
Error Detecting Code

We prove an accuracy threshold theorem for fault-tolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill, in which preparation circuits for ancilla states are protected by a concatenated error-detecting code and the preparation is aborted if an error is detected. The proof applies to independent stochastic noise but (in contrast to proofs of the quantum accuracy threshold theorem based on concatenated error-correcting codes) not to strongly-correlated adversarial noise. Our rigorously established lower bound on the accuracy threshold, $1.04\times 10^{-3}$, is well below Knill's numerical estimates.

scholarly journals The XZZX surface code

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
J. Pablo Bonilla Ataides ◽  
David K. Tuckett ◽  
Stephen D. Bartlett ◽  
Steven T. Flammia ◽  
Benjamin J. Brown
Keyword(s):  
Quantum Computation ◽  
High Performance ◽  
Quantum Computer ◽  
Fault Tolerant ◽  
Error Correcting Codes ◽  
The Common ◽  
Surface Code ◽  
Random Codes ◽  
Quantum Error ◽  
Relevant Range

AbstractPerforming large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.

Universal fault tolerant quantum computation on bilinear nearest neighbor arrays

2008 ◽  
Vol 8 (3&4) ◽  
pp. 330-344
Author(s):  
A.M. Stephens ◽  
A.G. Fowler ◽  
L.C.L. Hollenberg
Keyword(s):  
Lower Bound ◽  
Quantum Computation ◽  
Error Rate ◽  
Nearest Neighbor ◽  
Fault Tolerant ◽  
Quantum Code ◽  
Memory Errors ◽  
Memory Failure ◽  
Qubit Gate ◽  
Readout Error

Assuming an array that consists of two parallel lines of qubits and that permits only nearest neighbor interactions, we construct physical and logical circuitry to enable universal fault tolerant quantum computation under the $[[7,1,3]]$ quantum code. A rigorous lower bound to the fault tolerant threshold for this array is determined in a number of physical settings. Adversarial memory errors, two-qubit gate errors and readout errors are included in our analysis. In the setting where the physical memory failure rate is equal to one-tenth of the physical gate error rate, the physical readout error rate is equal to the physical gate error rate, and the duration of physical readout is ten times the duration of a physical gate, we obtain a lower bound to the asymptotic threshold of $1.96\times10^{-6}$.

scholarly journals ERROR THRESHOLD ESTIMATION BY MEANS OF THE [[7,1,3]] CSS QUANTUM CODE

2005 ◽  
Vol 03 (02) ◽  
pp. 371-393 ◽  
Author(s):  
P. J. SALAS ◽  
A. L. SANZ
Keyword(s):  
Quantum Computation ◽  
Error Probability ◽  
Fault Tolerant ◽  
Error Correcting Codes ◽  
Quantum Code ◽  
Threshold Estimation ◽  
Control Error ◽  
Threshold Conditions ◽  
Quantum Error ◽  
Depolarizing Channel

The states needed in quantum computation are extremely affected by decoherence. Several methods have been proposed to control error spreading. They use two main tools: fault-tolerant constructions and concatenated quantum error correcting codes. In this work, we estimate the threshold conditions necessary to make a long enough quantum computation. The [[7,1,3]] CSS quantum code together with the Shor method to measure the error syndrome is used. No concatenation is included. The decoherence is introduced by means of the depolarizing channel error model, obtaining several thresholds from the numerical simulation. Regarding the maintenance of a qubit stabilized in the memory, the error probability must be smaller than 2.9 × 10-5. In order to implement a one or two-qubit encoded gate in an effective fault-tolerant way, it is possible to choose an adequate non-encoded noisy gate if the memory error probability is smaller than 1.3 × 10-5. In addition, fulfilling this last condition permits us to assume a more efficient behavior compared to the equivalent non-encoded process.

Fault Tolerant Techniques for Finite Field Multipliers

2014 ◽  
Vol 573 ◽  
pp. 209-214
Author(s):  
B. Sargunam ◽  
R. Dhanasekaran
Keyword(s):  
Finite Field ◽  
Error Detection ◽  
Fault Tolerant ◽  
High Reliability ◽  
Concurrent Error Detection ◽  
Polynomial Representation ◽  
Finite Field Multiplier ◽  
On Line ◽  
Error Detecting ◽  
Modular Redundancy

The use of finite field multipliers in the critical applications like elliptic curve cryptography needs Concurrent Error Detection (CED) and correction at architectural level to provide high reliability. This paper discusses fault tolerant technique for polynomial representation based finite field multipliers. The detection and correction are done on-line. We use a combination of Double Modular Redundancy (DMR) and Concurrent Error Detection (CED) techniques. The fault tolerant finite field multiplier is coded in VHDL and simulated using Modelsim. Further, the proposed multiplier with fault tolerant capability is synthesized and results are analyzed with respect to area occupied, input and output pin counts and delay. Our technique, when compared with existing techniques, gives better performance. We show that our concurrent error detecting multiplier over GF(2m) requires less than 200% extra hardware, whereas with the traditional fault tolerant techniques, such as Triple Modular Redundancy (TMR), overhead is more than 200%.

scholarly journals Is error detection helpful on IBM 5Q chips?

2018 ◽  
Vol 18 (11&12) ◽  
pp. 949-964 ◽  
Author(s):  
Christophe Vuillot
Keyword(s):  
Quantum Computation ◽  
Error Detection ◽  
Fault Tolerant ◽  
Quantum Circuits ◽  
Average Improvement ◽  
Show Evidence ◽  
Fault Tolerant Design

This paper reports on experiments realized on several IBM~5Q chips which show evidence for the advantage of using error detection and fault-tolerant design of quantum circuits. We show an average improvement of the task of sampling from states that can be fault-tolerantly prepared in the [4,2,2] code, when using a fault-tolerant technique well suited to the layout of the chip. By showing that fault-tolerant quantum computation is already within our reach, the author hopes to encourage this approach.

Quantum accuracy threshold for concatenated distance-3 code

2006 ◽  
Vol 6 (2) ◽  
pp. 97-165 ◽  
Author(s):  
P. Aliferis ◽  
D. Gottesman ◽  
J. Preskill
Keyword(s):  
Lower Bound ◽  
Fault Tolerant ◽  
Combinatorial Analysis ◽  
Noise Model ◽  
Computer Assisted ◽  
Quantum Code ◽  
Stochastic Noise ◽  
Noise Models ◽  
New Criteria ◽  
Inductive Analysis

We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold $\varepsilon_0$. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, $\varepsilon_0 \ge 2.73\times 10^{-5}$ for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.

scholarly journals The boundaries and twist defects of the color code and their applications to topological quantum computation

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 101 ◽  
Author(s):  
Markus S. Kesselring ◽  
Fernando Pastawski ◽  
Jens Eisert ◽  
Benjamin J. Brown
Keyword(s):  
Quantum Computation ◽  
Domain Walls ◽  
Fault Tolerant ◽  
Error Correcting Codes ◽  
Two Dimensions ◽  
Abstract Theory ◽  
Color Code ◽  
Topological Quantum ◽  
Multiple Copies ◽  
Surface Code

The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, we discover a number of interesting new applications of the cataloged objects for quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum error-correcting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with bounded-weight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.

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