A Blueprint for Demonstrating Quantum Supremacy with Superconducting Qubits

Long coherence times and high fidelity control recently achieved in scalable superconducting circuits paved the way for the growing number of experimental studies of many-qubit quantum coherent phenomena in these devices. Albeit full implementation of quantum error correction and fault tolerant quantum computation remains a challenge the near term pre-error correction devices could allow new fundamental experiments despite inevitable accumulation of errors. One such open question foundational for quantum computing is achieving the so called quantum supremacy, an experimental demonstration of a computational task that takes polynomial time on the quantum computer whereas the best classical algorithm would require exponential time and/or resources. It is possible to formulate such a task for a quantum computer consisting of less than a 100 qubits. The computational task we consider is to provide approximate samples from a non-trivial quantum distribution. This is a generalization for the case of superconducting circuits of ideas behind boson sampling protocol for quantum optics introduced by Arkhipov and Aaronson. In this presentation we discuss a proof-of-principle demonstration of such a sampling task on a 9-qubit chain of superconducting gmon qubits developed by Google. We discuss theoretical analysis of the driven evolution of the device resulting in output approximating samples from a uniform distribution in the Hilbert space, a quantum chaotic state. We analyze quantum chaotic characteristics of the output of the circuit and the time required to generate a sufficiently complex quantum distribution. We demonstrate that the classical simulation of the sampling output requires exponential resources by connecting the task of calculating the output amplitudes to the sign problem of the Quantum Monte Carlo method. We also discuss the detailed theoretical modeling required to achieve high fidelity control and calibration of the multi-qubit unitary evolution in the device. We use a novel cross-entropy statistical metric as a figure of merit to verify the output and calibrate the device controls. Finally, we demonstrate the statistics of the wave function amplitudes generated on the 9-gmon chain and verify the quantum chaotic nature of the generated quantum distribution. This verifies the implementation of the quantum supremacy protocol

Open system quantum annealing in mean field models with exponential degeneracy

Real life quantum computers are inevitably affected by intrinsic noise resulting in dissipative non-unitary dynamics realized by these devices. We consider an open system quantum annealing algorithm optimized for a realistic analog quantum device which takes advantage of noise-induced thermalization and relies on incoherent quantum tunneling at finite temperature. We analyze the performance of this algorithm considering a p-spin model which allows for a mean field quasicalssical solution and at the same time demonstrates the 1st order phase transition and exponential degeneracy of states. We demonstrate that finite temperature effects introduced by the noise are particularly important for the dynamics in presence of the exponential degeneracy of metastable states. We determine the optimal regime of the open system quantum annealing algorithm for this model and find that it can outperform simulated annealing in a range of parameters.Comment: 11 pages, 5 figure

Non-universal weak antilocalization effect in cubic topological Kondo insulators

We study the quantum correction to conductivity on the surface of cubic topological Kondo insulators with multiple Dirac bands. We consider the model of time-reversal invariant disorder which induces the scattering of the electrons within the Dirac bands as well as between the bands. When only intraband scattering is present we find three long-range diffusion modes which lead to weak antilocalization correction to conductivity, which remains independent of the microscopic details such as Fermi velocities and relaxation times. Interband scattering gaps out two diffusion modes leaving only one long-range mode. We find that depending on the value of the phase coherence time, either three or only one long-range diffusion modes contribute to weak localization correction rendering the quantum correction to conductivity non-universal. We provide an interpretation for the results of the recent transport experiments on samarium hexaboride where weak antilocalization has been observed.Comment: 15 pages, 7 figure

Influence of Trotterization error on single-particle tunneling

Simulation of the single-particle tunneling problem by means of the Suzuki-Trotter approximation (STA) is analyzed. Considered is a particle hopping across a chain of sites in presence of a smooth position-dependent potential profile with several local minima that arrange a tunneling problem between the localized states in different minima. The STA error is found to manifest itself in three ways: i) perturbative energy shifts, ii) nonperturbartive renormalization of the tunneling rates, and iii) perturbative leakage of the total probability to other states. Generally, the first type of error is the most essential, as detuning of the tunneling resonance has to be compared with exponentially small tunneling rates. In absence of detuning (e.g. if the resonance is protected by symmetry), STA leads to exponential enhancement of the tunneling rates. The last type of error classifies the overall defect in the wave function and delineates the region of sufficiently weak distortion of the wave function due to STA. The conducted analysis confirms the naive criteria of applicability $\max\{T,P\}\ll\delta t^{-1}$ (with $T,P$ being the typical scales of kinetic and potential terms, respectively), while also revealing the structure of error and its behavior with system parameters. Analysis of the case of large Trotter step is also performed, with the main result being the reconstruction of low-energy spectrum due to coupling between states with energy difference close to $2\pi/\delta t$. The connection of the obtained results with rigorous upper error bounds on the STA error is discussed, with particular emphasis on why these rigorous bounds are not always saturated. We also point out that the proposed problem can be directly implemented on existing quantum devices [arXiv:2012.00921]. In particular, we give a detailed description of an experimental design that demonstrates the described physics.Comment: 38 pages, 12 figures. Changes: fixed several typos, replaced certain figures for better readabilit

Mechanism of Quantum Speedup in Novel Population Transfer Protocol for Binary Optimization Problems

We consider a novel quantum population transfer protocol to solve binary optimization problems that exploits quantum many-body dynamics in the delocalized regime. Hard optimization problems are characterized by energy landscape with a large number of local minima separated by large Hamming distances which scale with the problem size. This landscape gives rise to an interesting computational primitive: given an initial bit-string, we are to produce other bit-strings within certain narrow range of energies around the initial state. We consider a specific model we call "impurity band": a system of n qubits in a transverse field, where a number of bitstrings $M<<2^n$ selected at random are assigned random energies distributed in a narrow window of width $W<<1$ around the mean energy $-n$. We demonstrate the existence of the many-body delocalized regime in this model when the spectrum of the model splits into many-body minibands, and a typical eigenstate wave function is a superposition of peaks centered at a large number of local minima. The typical width of the minibands in energy determines the efficiency of the population transfer protocol. We demonstrate theoretically that the population transfer protocol achieves Grover type speedup in the unstructured impurity band model