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An introduction to quantum error correction and faulttolerant quantum computation
, 2009
"... Abstract. Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. ..."
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Abstract. Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers.
Level reduction and the quantum threshold theorem
 PH.D. THESIS, CALTECH, 2007, EPRINT ARXIV:QUANTPH/0703230
, 2007
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THRESHOLD ERROR RATES FOR THE TORIC AND SURFACE CODES
, 2009
"... The surface code scheme for quantum computation features a 2d array of nearestneighbor coupled qubits yet claims a threshold error rate approaching 1 % [1]. This result was obtained for the toric code, from which the surface code is derived, and surpasses all other known codes restricted to 2d neare ..."
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The surface code scheme for quantum computation features a 2d array of nearestneighbor coupled qubits yet claims a threshold error rate approaching 1 % [1]. This result was obtained for the toric code, from which the surface code is derived, and surpasses all other known codes restricted to 2d nearestneighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and surface codes, which is based on polynomialtime graph matching techniques and is efficiently implementable as the classical feedforward processing step in a real quantum computer. By direct simulation of this error correction scheme, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and nonideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of pth = 15.5 % under ideal syndrome extraction, and pth = 7.8×10 −3 for the nonideal case, in agreement with [1]. Simulations of the surface code indicate that the threshold is close to that of the toric code.
Abstract Running a Quantum Circuit at the Speed of Data
"... We analyze circuits for kernels from popular quantum computing applications, characterizing the hardware resources necessary to take ancilla preparation off the critical path. The result is a chip entirely dominated by ancilla generation circuits. To address this issue, we introduce optimized ancill ..."
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We analyze circuits for kernels from popular quantum computing applications, characterizing the hardware resources necessary to take ancilla preparation off the critical path. The result is a chip entirely dominated by ancilla generation circuits. To address this issue, we introduce optimized ancilla factories and analyze their structure and physical layout for ion trap technology. We introduce a new quantum computing architecture with highly concentrated dataonly regions surrounded by shared ancilla factories. The results are a reduced dependence on costly teleportation, more efficient distribution of generated ancillae and more than five times speedup over previous proposals. 1
permission. ESTIMATING THE RESOURCES FOR QUANTUM COMPUTATION WITH THE QuRE TOOLBOX
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Performance and Error Analysis of Knill’s Postselection Scheme in a TwoDimensional Architecture
"... Knill demonstrated a faulttolerant quantum computation scheme based on concatenated errordetecting codes and postselection with a simulated error threshold of 3 % over the depolarizing channel. We show how to use Knill’s postselection scheme in a practical twodimensional quantum architecture that ..."
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Knill demonstrated a faulttolerant quantum computation scheme based on concatenated errordetecting codes and postselection with a simulated error threshold of 3 % over the depolarizing channel. We show how to use Knill’s postselection scheme in a practical twodimensional quantum architecture that we designed with the goal to optimize the error correction properties, while satisfying important architectural constraints. In our 2D architecture, one logical qubit is embedded in a tile consisting of 5×5 physical qubits. The movement of these qubits is modeled as noisy SWAP gates and the only physical operations that are allowed are local one and twoqubit gates. We evaluate the practical properties of our design, such as its error threshold, and compare it to the concatenated BaconShor code and the concatenated Steane code. Assuming that all gates have the same error rates, we obtain a threshold of 3.06 × 10−4 in a local adversarial stochastic noise model, which is the highest known error threshold for concatenated codes in 2D. We also present a Monte Carlo simulation of the 2D architecture with depolarizing noise and we calculate a pseudothreshold of about 0.1%. With memory error rates onetenth of the worst gate error rates, the threshold for the adversarial noise model, and the pseudothreshold over depolarizing noise, are 4.06 × 10−4 and 0.2%, respectively. In a hypothetical technology where memory error rates are negligible, these thresholds can be further increased by shrinking the tiles into a 4 × 4 layout.
GRAPHICAL ALGORITHMS AND THRESHOLD ERROR RATES FOR THE 2D COLOUR CODE
, 907
"... Recent work on faulttolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1 % [1,2]. However, the 2d surface code requires the use of a complex state distillation procedure to achieve uni ..."
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Recent work on faulttolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1 % [1,2]. However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The colour code of [3] is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the colour code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25 % under errorfree syndrome extraction, while numerical simulations indicate it may be as high as 13.3%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.1%. 1
Abstract Resource Cost Derivation for Logical Quantum Circuit Descriptions
 in Proceedings of the 1st Workshop on Functional Programming Concepts in DSLs (FPCDSL 2013
, 2013
"... Resources that are necessary to operate a quantum computer (such as qubits) have significant costs. Thus, there is interest in finding ways to determine these costs for both existing and novel quantum algorithms. Information about these costs (and how they might vary under multiple parameters and c ..."
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Resources that are necessary to operate a quantum computer (such as qubits) have significant costs. Thus, there is interest in finding ways to determine these costs for both existing and novel quantum algorithms. Information about these costs (and how they might vary under multiple parameters and circumstances) can then be used to navigate tradeoffs and make optimizations within an algorithm implementation. We present a domainspecific language called QuIGL for describing logical quantum circuits; the QuIGL language has specialized features supporting the explicit annotation and automatic derivation of descriptions of the resource costs associated with each logical quantum circuit description (as well as any of its component procedures). We also present a formal framework for defining abstract transformations from QuIGL circuit descriptions into labelled, parameterized quantity expressions that can be used to compute exact counts or estimates of the cost of the circuit along chosen cost dimensions and for given input sizes. We demonstrate how this framework can be instantiated for calculating costs along specific dimensions (such as the number of qubits or the Tdepth of a logical quantum circuit).
1 HighLevel Interconnect Model for the Quantum Logic Array Architecture
, 2008
"... We summarize the main characteristics of the quantum logic array (QLA) architecture with a careful look at the key issues not described in the original conference publications: primarily, the teleportationbased logical interconnect. The design goal of the the quantum logic array architecture is to ..."
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We summarize the main characteristics of the quantum logic array (QLA) architecture with a careful look at the key issues not described in the original conference publications: primarily, the teleportationbased logical interconnect. The design goal of the the quantum logic array architecture is to illustrate a model for a largescale quantum architecture that solves the primary challenges of systemlevel reliability and data distribution over large distances. The QLA’s logical interconnect design, which employs the quantum repeater protocol, is in principle capable of supporting the communication requirements for applications as large as the factoring of a 2048bit number using Shor’s quantum factoring algorithm. Our physicallevel assumptions and architectural component validations are based on the trapped ion technology for implementing quantum computing.
QuRE: The Quantum Resource Estimator Toolbox
"... QuRE is a layout estimation tool that estimates the cost of practical implementations of quantum circuits in a variety of competing physical quantum technologies and with a variety of strategies for fault tolerant encoding. For each specified algorithm, QuRE estimates quantities such as number of ph ..."
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QuRE is a layout estimation tool that estimates the cost of practical implementations of quantum circuits in a variety of competing physical quantum technologies and with a variety of strategies for fault tolerant encoding. For each specified algorithm, QuRE estimates quantities such as number of physical qubits, execution time, probability of success of the computation, and physical gate counts for elementary quantum gate types of a specified technology. Out of the box, QuRE supports estimation for six physical quantum technologies, seven quantum algorithms, and with error correction using the Steane [1], [2], BaconShor [3], Knill [4] or surface [5], [6] error correction codes. Moreover, QuRE is extendable and can easily accommodate other choices. After describing QuRE, we use it to investigate the tradeoff between concatenated and surface error correction coding techniques, demonstrating the existence of a crossover point for the Ground State Estimation Algorithm [7]. I.