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arXiv:2301.03725 · Physical Review A

Qubit recycling and the path-counting problem

Zijian Song, Isaac H. Kim

Published Oct 18, 2023• 1 citation
arXivHTMLPDF

Abstract

Recently, it was shown that the qudits used in circuits with a convolutional form (e.g., matrix product states and the multiscale entanglement renormalization Ansatz) can be reset unitarily [Anikeeva et al., Phys. Rev. A 103, 042613 (2021)], even without measurement. We analyze the fidelity of this protocol for a family of quantum circuits that interpolates between such circuits and local quantum circuits, averaged over Haar-random gates. We establish a connection between this problem and counting directed paths on a graph, which is determined by the shape of the quantum circuit. This connection leads to an exact expression for the fidelity of the protocol for the entire family that interpolates between a convolutional circuit and a random quantum circuit. For convolutional circuits with a constant window size, the rate of convergence to unit fidelity is shown to be $\frac{{q}^{2}}{{q}^{2}+1}$, independent of the window size, where $q$ is the local qudit dimension. Since most applications of convolutional circuits use constant-size windows, our result suggests that the unitary reset protocol will likely work well in such a regime. We also derive two extra results in the convolutional limit, which may be of independent interest. First, we derive exact expressions for the correlations between reset qudits and show that they decays exponentially in the distance. Second, we derive an expression for the fidelity in the presence of noise, expressed in terms of the quantities that define the property of the channel, such as the entanglement fidelity.

DOI
10.1103/physreva.108.042416

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Present-day quantum processors are open systems in which dissipation and decoherence degrade the information carried by a quantum state as it propagates through a circuit. We study this degradation in dissipative random quantum circuits, modeling each two-qubit gate as a diluted unitary that interpolates between the intended unitary operation and a random dissipative quantum channel. Using the fidelity between the ideal and noisy output states as a diagnostic, we show that sufficiently random unitary gates or dissipative Kraus operators lead to a universal decay of the average fidelity. This decay is determined only by the dissipation strength, system size, and circuit depth, while microscopic details of the gates and noise, including the Kraus rank, appear only in subleading corrections to higher moments. We derive a closed-form expression for the average fidelity in terms of three physically meaningful error parameters, dissipation strength, coherent two-qubit gate-error strength, and connectivity-error probability. Finally, we identify the cases when dissipative effects can be reproduced by an unitary-noise model and when dissipation-induced decoherence remains distinguishable from coherent noise at the level of the average fidelity.
We consider random quantum circuits (RQC) on arbitrary connected graphs whose edges determine the allowed 222-qudit interactions. Prior work has established that such nnn-qudit circuits with local dimension qqq on 1D, complete, and DDD-dimensional graphs form approximate unitary designs, that is, they generate unitaries from distributions close to the Haar measure on the unitary group U(qn)U(q^n)U(qn) after polynomially many gates. Here, we extend those results by proving that RQCs comprised of O(poly(n,k))O(\mathrm{poly}(n,k))O(poly(n,k)) gates on a wide class of graphs form approximate unitary kkk-designs. We prove that RQCs on graphs with spanning trees of bounded degree and height form kkk-designs after O(∣E∣n poly(k))O(|E|n\,\mathrm{poly}(k))O(∣E∣npoly(k)) gates, where ∣E∣|E|∣E∣ is the number of edges in the graph. Furthermore, we identify larger classes of graphs for which RQCs generate approximate designs in polynomial circuit size. For k≤4k \leq 4k≤4, we show that RQCs on graphs of certain maximum degrees form designs after O(∣E∣n)O(|E|n)O(∣E∣n) gates, providing explicit constants. We determine our circuit size bounds from the spectral gaps of local Hamiltonians. To that end, we extend the finite-size (or Knabe) method for bounding gaps of frustration-free Hamiltonians on regular graphs to arbitrary connected graphs. We further introduce a new method based on the Detectability Lemma for determining the spectral gaps of Hamiltonians on arbitrary graphs. Our methods have wider applicability as the first method provides a succinct alternative proof of [Commun. Math. Phys. 291, 257 (2009)] and the second method proves that RQCs on any connected architecture form approximate designs in quasi-polynomial circuit size.
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We study a random unitary quantum circuit with only reset channels, which has high feasibility for real quantum devices. In particular, we investigate the many-body statistical physics properties, ``reset-induced'' entanglement phase transitions comparing the classical statistical picture in the large ``ddd'' limit of qudits. In the property of the reset-induced phase transition the parameter of qudit ddd is essential. That is, the transition properties induced by the reset channel significantly depend on ddd. We numerically elucidate this statement employing efficient stabilizer circuit simulations for d=2d=2d=2. Specifically, critical fluctuations that grow with system size are observed near the transition point, and the finite-size scaling analyses yield data collapse consistent with a continuous mixed-state phase transition.} This behavior differs from expectations based on the classical statistical mapping in the large-ddd limit.
In this thesis, we are concerned with the problem of characterizing noise associated with implementations of quantum circuits. We first explore the notion of error rates of quantum circuits and argue that the semantic distinction between process fidelity as “average error rate” and diamond distance as “worst-case error rate” is wrong. To this end, we analyze their worst-case and average analogies, and show they are proportional to their original measures. We show that the non-unital contributions to the diamond distance are negligible and the discrepancy between the process fidelity and diamond distance is primarily due to unitary errors. \n \nWe then turn to a new analysis of cycle benchmarking, a randomized benchmarking-like protocol to estimate the process fidelity of a cycle as engineered by randomized compiling. Using this approach, we prove that gate-dependent noise on the randomized gates is de- scribed by a single perturbation term that decays rapidly as long as the implementation is close to a representation. We also comment on how the protocol can be extended to qudits, what the cycle benchmarking decay actually measures, and how our analysis is amenable to the Fourier analysis of randomized benchmarking. We end with a discussion of how the gate-dependent cycle benchmarking process fidelity relates to gate-dependent randomized compiling.
Abstract We investigate the fidelity of Haar random bipartite pure states from a fixed reference quantum state and their bipartite entanglement. By plotting the fidelity and entanglement on perpendicular axes, we observe that the resulting plots exhibit non-uniform distributions. The distribution depends on the entanglement of the fixed reference quantum state used to quantify the fidelity of the random pure bipartite states. We find that the average fidelity of typical random pure bipartite qubits within a narrow entanglement range with respect to a randomly chosen fixed bipartite qubit is 1 4 . Extending our study to higher dimensional bipartite qudits, we find that the average fidelity of typical random pure bipartite qudits with respect to a randomly chosen fixed bipartite qudit remains constant within a narrow entanglement range. The values of these constants are 1 d 2 , with d being the dimension of the local Hilbert space of the bipartite qudit system, suggesting a consistent relationship between entanglement and fidelity across different dimensions. The probability distribution functions of fidelity with respect to a product state are analytically studied and used as a reference for the benchmarking of distributed quantum computing devices.