Parameters of Pseudo-Random Quantum Circuits

arXiv (Cornell University), 2022

We study the approximate state preparation problem on noisy intermediate-scale quantum (NISQ) computers by applying a genetic algorithm to generate quantum circuits for state preparation. The algorithm can account for the specific characteristics of the physical machine in the evaluation of circuits, such as the native gate set and qubit connectivity. We use our genetic algorithm to optimize the circuits provided by the low-rank state preparation algorithm introduced by Araujo et al., and find substantial improvements to the fidelity in preparing Haar random states with a limited number of CNOT gates. Moreover, we observe that already for a 5-qubit quantum processor with limited qubit connectivity and significant noise levels (IBM Falcon 5T), the maximal fidelity for Haar random states is achieved by a short approximate state preparation circuit instead of the exact preparation circuit. We also present a theoretical analysis of approximate state preparation circuit complexity to motivate our findings. Our genetic algorithm for quantum circuit discovery is freely available at https://github.com/beratyenilen/qc-ga.

2015

The Journal of Physical Chemistry Letters

The first achievement of quantum supremacy has been claimed recently by Google for the random quantum circuit benchmark with 53 superconducting qubits. Here, we analyze the randomness of Google's quantum random-bit sampling. The heat maps of Google's random bit-strings show stripe patterns at specific qubits in contrast to the Haar-measure or classical random-bit strings. Google's data contains more bit 0 than bit 1, i.e., about 2.8% difference, and fail to pass the NIST random number tests, while the Haar-measure or classical random-bit samples pass. Their difference is also illustrated by the Marchenko-Pastur distribution and the Girko circular law of random matrices of random bit-strings. The calculation of the Wasserstein distances shows that Google's random bit-strings are farther away from the Haar-measure random bit-strings than the classical random bit-strings. Our results imply that random matrices and the Wasserstein distance could be new tools for analyzing the performance of quantum computers.

npj Quantum Information

Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any random unitaries has consequences on their complexity, expressibility, and trainability. To study this property of random circuits, we develop numerical protocols for estimating the frame potential, the distance between a given ensemble and the exact randomness. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high-performing using CPU and GPU parallelism. We study 1. local and parallel random circuits to verify the linear growth in complexity as stated by the Brown–Susskind conjecture, and; 2. hardware-efficient ansätze to shed light on its expressibility and the barren plateau problem in the context of variational algorithms. Our work shows that large-scale tensor network simulations could provide importan...

2004

Current quantum computing hardware is unable to sustain quantum coherent operations for more than a handful of gate operations. Consequently, if near-term experimental milestones, such as synthesizing arbitrary entangled states, or performing fault-tolerant operations, are to be met, it will be necessary to minimize the number of elementary quantum gates used.

Physical Review A

Random quantum sampling, a task to sample bit-strings from a random quantum circuit, is considered one of suitable benchmark tasks to demonstrate the outperformance of quantum computers even with noisy qubits. Recently, random quantum sampling was performed on the Sycamore quantum processor with 53 qubits [Nature 574, 505 (2019)] and on the Zuchongzhi quantum processor with 56 qubits [Phys. Rev. Lett. 127, 180501 (2021)]. Here, we analyze and compare statistical properties of the outputs of random quantum sampling by Sycamore and Zuchongzhi. Using the Marchenko-Pastur law and the Wasssertein distances, we find that quantum random sampling of Zuchongzhi is more closer to classical uniform random sampling than those of Sycamore. Some Zuchongzhi's bit-strings pass the random number tests while both Sycamore and Zuchongzhi show similar patterns in heatmaps of bit-strings. It is shown that statistical properties of both random quantum samples change little as the depth of random quantum circuits increases. Our findings raise a question about computational reliability of noisy quantum processors that could produce statistically different outputs for the same random quantum sampling task.

Quantum, 2023

For a random set S ⊂ U (d) of quantum gates we provide bounds on the probability that S forms a δ-approximate t-design. In particular we have found that for S drawn from an exact t-design the probability that it forms a δapproximate t-design satisfies the inequality , where D t is a sum over dimensions of unique irreducible representations appearing in the decomposition of U → U ⊗t ⊗ Ū ⊗t . We use our results to show that to obtain a δ-approximate t-design with probability P one needs O(δ -2 (t log(d) -log(1 -P ))) many random gates. We also analyze how δ concentrates around its expected value Eδ for random S. Our results are valid for both symmetric and non-symmetric sets of gates.

arXiv (Cornell University), 2022

Random circuit sampling, the task to sample bit strings from a random unitary operator, has been performed to demonstrate quantum advantage on the Sycamore quantum processor with 53 qubits and on the Zuchongzhi quantum processor with 56 and 61 qubits. Recently, it has been claimed that classical computers using tensor network simulation could catch on current noisy quantum processors for random circuit sampling. While the linear cross entropy benchmark fidelity is used to certify all these claims, it may not capture in detail statistical properties of outputs. Here, we compare the bit strings sampled from classical computers using tensor network simulation by Pan et al.

arXiv (Cornell University), 2022

For a Haar random set S ⊂ U (d) of quantum gates we consider the uniform measure ν S whose support is given by S. The measure ν S can be regarded as a δ(ν S , t)-approximate t-design, t ∈ Z + . We propose a random matrix model that aims to describe the probability distribution of δ(ν S , t) for any t. Our model is given by a block diagonal matrix whose blocks are independent, given by Gaussian or Ginibre ensembles, and their number, size and type is determined by t. We prove that, the operator norm of this matrix, δ(t), is the random variable to which |S|δ(ν S , t) converges in distribution when the number of elements in S grows to infinity. Moreover, we characterize our model giving explicit bounds on the tail probabilities P(δ(t) > 2 + ϵ), for any ϵ > 0. We also show that our model satisfies the so-called spectral gap conjecture, i.e. we prove that with the probability 1 there is t ∈ Z + such that sup k∈Z + δ(k) = δ(t). Numerical simulations give convincing evidence that the proposed model is actually almost exact for any cardinality of S. The heuristic explanation of this phenomenon, that we provide, leads us to conjecture that the tail probabilities P( √ Sδ(ν S , t) > 2 + ϵ) are bounded from above by the tail probabilities P(δ(t) > 2 + ϵ) of our random matrix model. In particular our conjecture implies that a Haar random set S ⊂ U (d) satisfies the spectral gap conjecture with the probability 1. 1 6 / log(d)). Moreover, in it was shown that random circuits built form Haar-random 2-qubit gates acting (according to the specified layout)

arXiv: Quantum Physics, 2019

We give three new algorithms for efficient in-place estimation, without using ancilla qubits, of average fidelity of a quantum logic gate acting on a d-dimensional system using much fewer random bits than what was known so far. Previous approaches for efficient estimation of average gate fidelity replaced Haar random unitaries in the naive estimation algorithm by approximate unitary 2-designs, and sampled them uniformly and independently. In contrast, in our first algorithm we sample the unitaries of the approximate unitary 2-design uniformly using a limited independence pseudorandom generator, a powerful tool from derandomisation theory. This algorithm uses the same number of basic operations as previous efficient algorithms but much fewer number of random bits. Reducing the requirement of classical random bits increases the reliability of estimation as often, high quality random bits are an expensive computational resource. Our second efficient algorithm, based on a 4-quantum tens...