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Quantum binary field inversion: improved circuit depth via choice of basis representation
 Quantum Information & Computation
, 2013
"... Finite fields of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal ba ..."
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Finite fields of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of ‘ghostbit basis’ representations can be used to implement inverters with a quantum circuit of depth O(m log(m)). To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the ItohTsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields F2m based on an elementary gate set that is useful for faulttolerant implementation. 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).
NONLOCALITY AS A BENCHMARK FOR UNIVERSAL QUANTUM COMPUTATION IN ISING ANYON TOPOLOGICAL QUANTUM COMPUTERS
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Systematic distillation of composite Fibonacci anyons using one mobile quasiparticle
"... A topological quantum computer should allow intrinsically faulttolerant quantum computation, but there remains uncertainty about how such a computer can be implemented. It is known that topological quantum computation can be implemented with limited quasiparticle braiding capabilities, in fact usi ..."
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A topological quantum computer should allow intrinsically faulttolerant quantum computation, but there remains uncertainty about how such a computer can be implemented. It is known that topological quantum computation can be implemented with limited quasiparticle braiding capabilities, in fact using only a single mobile quasiparticle, if the system can be properly initialized by measurements. It is also known that measurements alone suffice without any braiding, provided that the measurement devices can be dynamically created and modified. We study a model in which both measurement and braiding capabilities are limited. Given the ability to pull nontrivial Fibonacci anyon pairs from the vacuum with a certain success probability, we show how to simulate universal quantum computation by braiding one quasiparticle and with only one measurement, to read out the result. The difficulty lies in initializing the system. We give a systematic construction of a family of braid sequences that initialize to arbitrary accuracy nontrivial composite anyons. Instead of using the SolovayKitaev theorem, the sequences are based on a quantum algorithm for convergent search. 1