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35
Improved simulation of stabilizer circuits
 Phys. Rev. Lett
"... The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we ..."
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Cited by 65 (6 self)
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The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freelyavailable program called CHP (CNOTHadamardPhase), which can handle thousands of qubits easily. • We show that the problem of simulating stabilizer circuits is complete for the classical complexity class ⊕L, which means that stabilizer circuits are probably not even universal for classical computation. • We give efficient algorithms for computing the inner product between two stabilizer states, putting any nqubit stabilizer circuit into a “canonical form ” that requires at most O ( n 2 /log n) gates, and other useful tasks. • We extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensorproduct initial states but containing only a limited number of measurements. 1
Quantum walks: a comprehensive review
, 2012
"... Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting ..."
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Cited by 21 (0 self)
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Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete and continuoustime quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discretetime quantum walks. Furthermore, we have reviewed several algorithms based on both discrete and continuoustime quantum walks as well as a most important result: the computational universality of both continuous and discretetime quantum walks.
Graphbased Simulation of Quantum Computation in the Density Matrix Representation
 Journal of Quantum Information and Computation
, 2005
"... Quantummechanical phenomena are playing an increasing role in information processing, as transistor sizes approach the nanometer level, and quantum circuits and data encoding methods appear in the securest forms of communication. Simulating such phenomena efficiently is exceedingly difficult becaus ..."
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Cited by 17 (3 self)
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Quantummechanical phenomena are playing an increasing role in information processing, as transistor sizes approach the nanometer level, and quantum circuits and data encoding methods appear in the securest forms of communication. Simulating such phenomena efficiently is exceedingly difficult because of the vast size of the quantum state space involved. A major complication is caused by errors (noise) due to unwanted interactions between the quantum states and the environment. Consequently, simulating quantum circuits and their associated errors using the density matrix representation is potentially significant in many applications, but is well beyond the computational abilities of most classical simulation techniques in both time and memory resources. The size of a density matrix grows exponentially with the number of qubits simulated, rendering arraybased simulation techniques that explicitly store the density matrix intractable. In this work, we propose a new technique aimed at efficiently simulating quantum circuits that are subject to errors. In particular, we describe new graphbased algorithms implemented in the simulator QuIDDPro/D. While previously reported graphbased simulators operate in terms of the statevector representation, these new algorithms use the density matrix representation. To gauge the improvements offered by QuIDDPro/D, we compare its simulation performance with an optimized arraybased simulator called QCSim. Empirical results, generated by both simulators on a set of quantum circuit benchmarks involving error correction, reversible logic, communication, and quantum search, show that the graphbased approach far outperforms the arraybased approach.
Checking Equivalence of Quantum Circuits and States
, 2007
"... Quantum computing promises exponential speedups for important simulation and optimization problems. It also poses new CAD problems that are similar to, but more challenging, than the related problems in classical (nonquantum) CAD, such as determining if two states or circuits are functionally equiv ..."
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Cited by 17 (0 self)
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Quantum computing promises exponential speedups for important simulation and optimization problems. It also poses new CAD problems that are similar to, but more challenging, than the related problems in classical (nonquantum) CAD, such as determining if two states or circuits are functionally equivalent. While differences in classical states are easy to detect, quantum states, which are represented by complexvalued vectors, exhibit subtle differences leading to several notions of equivalence. This provides flexibility in optimizing quantum circuits, but leads to difficult new equivalencechecking issues for simulation and synthesis. We identify several different equivalencechecking problems and present algorithms for practical benchmarks, including quantum communication and search circuits, which are shown to be very fast and robust for hundreds of qubits.
A Fault Tolerant, Area Efficient Architecture for Shor’s Factoring Algorithm
"... We optimize the area and latency of Shor’s factoring while simultaneously improving fault tolerance through: (1) balancing the use of ancilla generators, (2) aggressive optimization of error correction, and (3) tuning the core adder circuits. Our custom CAD flow produces detailed layouts of the phys ..."
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Cited by 14 (3 self)
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We optimize the area and latency of Shor’s factoring while simultaneously improving fault tolerance through: (1) balancing the use of ancilla generators, (2) aggressive optimization of error correction, and (3) tuning the core adder circuits. Our custom CAD flow produces detailed layouts of the physical components and utilizes simulation to analyze circuits in terms of area, latency, and success probability. We introduce a metric, called ADCR, which is the probabilistic equivalent of the classic AreaDelay product. Our error correction optimization can reduce ADCR by an order of magnitude or more. Contrary to conventional wisdom, we show that the area of an optimized quantum circuit is not dominated exclusively by error correction. Further, our adder evaluation shows that quantum carrylookahead adders (QCLA) beat ripplecarry adders in ADCR, despite being larger and more complex. We conclude with what we believe is one of most accurate estimates of the area and latency required for 1024bit Shor’s factorization: 7659 mm 2 for the smallest circuit and 6 × 10 8 seconds for the fastest circuit.
Probabilistic transfer matrices in symbolic reliability analysis of logic circuits
 ACM Transactions on Design Automation of Electronic Systems
"... We propose the probabilistic transfer matrix (PTM) framework to capture nondeterministic behavior in logic circuits. PTMs provide a concise description of both normal and faulty behavior, and are wellsuited to reliability and error susceptibility calculations. A few simple composition rules based o ..."
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Cited by 13 (2 self)
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We propose the probabilistic transfer matrix (PTM) framework to capture nondeterministic behavior in logic circuits. PTMs provide a concise description of both normal and faulty behavior, and are wellsuited to reliability and error susceptibility calculations. A few simple composition rules based on connectivity can be used to recursively build larger PTMs (representing entire logic circuits) from smaller gate PTMs. PTMs for gates in series are combined using matrix multiplication, and PTMs for gates in parallel are combined using the tensor product operation. PTMs can accurately calculate joint output probabilities in the presence of reconvergent fanout and inseparable joint input distributions. To improve computational efficiency, we encode PTMs as algebraic decision diagrams (ADDs). We also develop equivalent ADD algorithms for newly defined matrix operations such as eliminate variables and eliminate redundant variables, which aid in the numerical computation of circuit PTMs. We use PTMs to evaluate circuit reliability and derive polynomial approximations for circuit error probabilities in terms of gate error probabilities. PTMs can also analyze the effects of logic and electrical masking on error mitigation. We show that ignoring logic masking can overestimate errors by an order of magnitude. We incorporate electrical masking by computing error attenuation probabilities, based on analytical models, into an extended PTM
Lukac M.: ‘Test Generation and Fault Localization for Quantum Circuits
 Proceedings of 35 th International Symposium on MultipleValued Logic
"... It is believed that quantum computing will begin to have a practical impact in industry around year 2010. We propose an approach to test generation and fault localization for a wide category of fault models. While in general we follow the methods used in test of standard circuits, there are two sign ..."
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Cited by 8 (0 self)
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It is believed that quantum computing will begin to have a practical impact in industry around year 2010. We propose an approach to test generation and fault localization for a wide category of fault models. While in general we follow the methods used in test of standard circuits, there are two significant differences: (2) we use both deterministic and probabilistic tests to detect faults, (2) we use special measurement gates to determine the internal states. A Fault Table is created that includes probabilistic information. “Probabilistic set covering” and “probabilistic adaptive trees ” that generalize those known in standard circuits, are next used. 1.
Debugging of toffoli networks
 In Design, Automation and Test in Europe
, 2009
"... Abstract—Intensive research is performed to find postCMOS technologies. A very promising direction based on reversible logic are quantum computers. While in the domain of reversible logic synthesis, testing, and verification have been investigated, debugging of reversible circuits has not yet been ..."
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Cited by 8 (8 self)
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Abstract—Intensive research is performed to find postCMOS technologies. A very promising direction based on reversible logic are quantum computers. While in the domain of reversible logic synthesis, testing, and verification have been investigated, debugging of reversible circuits has not yet been considered. The goal of debugging is to determine gates of an erroneous circuit that explain the observed incorrect behavior. In this paper we propose the first approach for automatic debugging of reversible Toffoli networks. Our method uses a formulation for the debugging problem based on Boolean satisfiability. We show the differences to classical (irreversible) debugging and present theoretical results. These are used to speedup the debugging approach as well as to improve the resulting quality. Our method is able to find and to correct single errors automatically. I.
A Quantum CAD Accelerator Based on Grover’s Algorithm for Finding the Minimum Fixed Polarity ReedMuller Form
"... We describe the use of Grover’s algorithm as implemented in a quantum logic circuit that produces a solution for a classical switching circuit design problem. The particular application described here is to determine a Fixed Polarity ReedMuller (FPRM) form that satisfies a threshold value constrain ..."
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Cited by 4 (0 self)
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We describe the use of Grover’s algorithm as implemented in a quantum logic circuit that produces a solution for a classical switching circuit design problem. The particular application described here is to determine a Fixed Polarity ReedMuller (FPRM) form that satisfies a threshold value constraint, thus we find a particular FPRM form among all 2 n FPRM forms that has a number of terms less than or equal to the threshold value. Grover’s algorithm is implemented in a quantum logic circuit that also contains a subcircuit that expresses all possible FPRM solutions of a given function. This approach illustrates how fast transforms as known from spectral theory can be combined with quantum computing as a part of an oracle. 1.
HighPerformance QuIDDbased Simulation of Quantum Circuits
 Proc. Design Automation and Test in Europe Conf., Paris
, 2004
"... Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with the number of qubits. It has been shown experimentally that the QuIDD (Quantum Information Decision Diagram) datastructure ..."
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Cited by 4 (1 self)
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Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with the number of qubits. It has been shown experimentally that the QuIDD (Quantum Information Decision Diagram) datastructure greatly facilitates simulations using memory and runtime that are polynomial in the number of qubits. In this paper, we present a complexity analysis which formally describes this class of matrices and vectors. We also present an improved implementation of QuIDDs which can simulate Grover's algorithm for quantum search with the asymptotic runtime complexity of an ideal quantum computer up to negligible overhead.