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A Parallel Simulator for Quantum Fault Tolerance in the Presence of Correlated Errors

Dan C. Marinescu, Gabriela M. Marinescu

2008
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2008 22nd Workshop on Principles of Advanced and Distributed Simulation
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The only realistic means to assess the reliability of a fault tolerant quantum circuit is through simulation. Quantum circuit simulators use Monte Carlo (MC) strategy to randomly sample possible error scenarios using a random number generator. Monte Carlo simulation can be conducted in parallel, but it requires a large amount of CPU cycles and a fair amount of time. For example, it is reported that a simulation of a circuit with two logical qubits required 8 (eight) days of simulation time on a
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... single system with 1 GB of main memory. This is quite understandable; if a rare event e.g., an uncorrectable quantum error, occurs once every ν runs then, in order to generate enough samples, the number of simulation runs should be several orders of magnitudes larger than ν; for example, if the probability of an uncorrectable error is 10 −6 the number of runs should be of the order of 10 9 . Recently, a group from Princeton proposed a new methodology based upon a probability tree that tracks the physical qubit errors. The authors report a speedup of four orders of magnitude versus the MC simulation for the same circuit and the same level of accuracy. Unfortunately, the memory requirements of this method are very steep and the scalability of the method is rather poor due to the combinatorial approach; a simulation of a fault-tolerant Toffoli gate with the [[7, 1, 3]] single-error correcting Hamming code and 22 non-failing error states per code block requires 1 GB of memory while a simulation of the [[23, 1, 5]] double-error correcting Golay code and 1954 non-failing error states per code block was estimated to require nearly 90 GB of memory. All quantum circuit simulators assume a Markovian quantum error model. Yet physical quantum devices are affected by space-and time-correlated errors. Correlated quantum errors occur with a lower probability than uncorrelated ones; the probability of system crashes when we use either traditional quantum error-correcting codes, or codes specially designed to deal with time-correlated errors is even lower than the one for Markovian noise models. The simulator we are currently developing uses a combi- * natorial model and the simulation is carried out in parallel. The simulator is driven by a component which triggers noise events based upon the noise model for a particular physical implementation of quantum gates. The qubits are grouped together based upon the topology of the circuit into qubit sets. An error map encodes strings of Pauli operators and the associated probability; I, X, Y, Z stand for the Pauli operators σ I , σ X , σ Y , σ Z , respectively. A correlation map allows us to simulate time-and space-correlated errors. The probability tree is rooted in some initial state and it is expanded as a result of error events. We assume that at each time step a freely evolving physical qubit does not change state with probability 1 − p or undergoes rotation by Pauli operators σ X , σ Y , σ Z with probability p/3 each. As the computation progresses, several qubit sets, whose members are either applied to the same gate, or involved in the elementary step of the computation, are merged together; correlated errors also can trigger merging of qubit sets. Qubit sets are split whenever their members do no longer act in concert. There is a tradeoff between the accuracy of the simulation and the computational effort and time. We can perform a lossy merge and ignore states with probability below a merge threshold and limit the tree expansion. Merging qubit sets, while trivial when the probability tree is monolithic, is likely to generate a fair amount of communication when the probability tree is distributed across the nodes of a cluster. To improve the scalability of the combinatorial approach we have to minimize communication in the distributed simulation environment. To do so we investigate multiple paths including: (i) optimal data partitioning strategies; (ii) data replication; (iii) strategies to limit the expansion of the probability tree, e.g., threshold methods; (iv) an efficient state and error encoding scheme; and (v) election of a suitable architecture of the system we plan to run the simulation on. 22nd Workshop on Principles of Advanced and Distributed Simulation

doi:10.1109/pads.2008.34
dblp:conf/pads/MarinescuM08
fatcat:vlcxvhbi7reuznaxisz5kez2jm