 Award ID(s):
 1917383
 Publication Date:
 NSFPAR ID:
 10189442
 Journal Name:
 nternational Conference on Reversible Computation (RC), Springer LNCS
 Volume:
 11497
 Sponsoring Org:
 National Science Foundation
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A basic question in the theory of faulttolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci TuraevViro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggestmore »

Abstract The road to computing on quantum devices has been accelerated by the promises that come from using Shor’s algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and lack of robust error correction schemes. Here we explore a promising, alternative method for prime factorization that uses wellestablished techniques from variational imaginary time evolution. We create a Hamiltonian whose ground state encodes the solution to the problem and use variational techniques to evolve a state iteratively towards these prime factors. We show that the number of circuits evaluated in each iteration scales as
, where$$O(n^{5}d)$$ $O\left({n}^{5}d\right)$n is the bitlength of the number to be factorized andd is the depth of the circuit. We use a single layer of entangling gates to factorize 36 numbers represented using 7, 8, and 9qubit Hamiltonians. We also verify the method’s performance by implementing it on the IBMQ Lima hardware to factorize 55, 65, 77 and 91 which are greater than the largest number (21) to have been factorized on IBMQ hardware. 
Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy ( P H ) does not collapse, a stronger version of the statement that P ≠ N P , which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing finegrained versions of the noncollapse conjecture. Our first two conjectures poly3NSETH( a ) and perintNSETH( b ) take specific classical counting problems related to the number of zeros of a degree3 polynomial in n variables over F 2 or the permanent of an n × n integervalued matrix, and assert that any nondeterministic algorithm that solves them requires 2 c nmore »

The current phase of quantum computing is in the Noisy IntermediateScale Quantum (NISQ) era. On NISQ devices, twoqubit gates such as CNOTs are much noisier than singlequbit gates, so it is essential to minimize their count. Quantum circuit synthesis is a process of decomposing an arbitrary unitary into a sequence of quantum gates, and can be used as an optimization tool to produce shorter circuits to improve overall circuit fidelity. However, the timetosolution of synthesis grows exponentially with the number of qubits. As a result, synthesis is intractable for circuits on a large qubit scale. In this paper, we propose a hierarchical, blockbyblock optimization framework, QGo, for quantum circuit optimization. Our approach allows an exponential cost optimization to scale to large circuits. QGo uses a combination of partitioning and synthesis: 1) partition the circuit into a sequence of independent circuit blocks; 2) regenerate and optimize each block using quantum synthesis; and 3) recompose the final circuit by stitching all the blocks together. We perform our analysis and show the fidelity improvements in three different regimes: smallsize circuits on real devices, mediumsize circuits on noisy simulations, and largesize circuits on analytical models. Our technique can be applied after existing optimizations tomore »

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