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1
NAKATA, YOSHIFUMI, and MIO MURAO. "DIAGONAL-UNITARY 2-DESIGN AND THEIR IMPLEMENTATIONS BY QUANTUM CIRCUITS." International Journal of Quantum Information 11, no. 07 (October 2013): 1350062. http://dx.doi.org/10.1142/s0219749913500627.
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We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitaryt-designs and present two quantum circuits that implement diagonal-unitary 2-design with the computational basis in N-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary 2-design after applying the gates on all pairs of qubits. The number of required gates is N(N - 1)/2. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact 2-design, but achieves an ϵ-approximate 2-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most O(N2(N + log 1/∊)). We also provide an application of the circuits, a protocol of generating an exact 2-design of random states by combining the circuits with a simple classical procedure requiring O(N) random classical bits.
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Rampp, Michael A., and Pieter W. Claeys. "Hayden-Preskill recovery in chaotic and integrable unitary circuit dynamics." Quantum 8 (August 8, 2024): 1434. http://dx.doi.org/10.22331/q-2024-08-08-1434.
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The Hayden-Preskill protocol probes the capability of information recovery from local subsystems after unitary dynamics. As such it resolves the capability of quantum many-body systems to dynamically implement a quantum error-correcting code. The transition to coding behavior has been mostly discussed using effective approaches, such as entanglement membrane theory. Here, we present exact results on the use of Hayden-Preskill recovery as a dynamical probe of scrambling in local quantum many-body systems. We investigate certain classes of unitary circuit models, both structured Floquet (dual-unitary) and Haar-random circuits. We discuss different dynamical signatures corresponding to information transport or scrambling, respectively, that go beyond effective approaches. Surprisingly, certain chaotic circuits transport information with perfect fidelity. In integrable dual-unitary circuits, we relate the information transmission to the propagation and scattering of quasiparticles. Using numerical and analytical insights, we argue that the qualitative features of information recovery extend away from these solvable points. Our results suggest that information recovery protocols can serve to distinguish chaotic and integrable behavior, and that they are sensitive to characteristic dynamical features, such as long-lived quasiparticles or dual-unitarity.
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Claeys, Pieter W., and Austen Lamacraft. "Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics." Quantum 6 (June 15, 2022): 738. http://dx.doi.org/10.22331/q-2022-06-15-738.
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Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.
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Turkeshi, Xhek, and Piotr Sierant. "Hilbert Space Delocalization under Random Unitary Circuits." Entropy 26, no. 6 (May 29, 2024): 471. http://dx.doi.org/10.3390/e26060471.
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The unitary dynamics of a quantum system initialized in a selected basis state yield, generically, a state that is a superposition of all the basis states. This process, associated with the quantum information scrambling and intimately tied to the resource theory of coherence, may be viewed as a gradual delocalization of the system’s state in the Hilbert space. This work analyzes the Hilbert space delocalization under the dynamics of random quantum circuits, which serve as a minimal model of the chaotic dynamics of quantum many-body systems. We employ analytical methods based on the replica trick and Weingarten calculus to investigate the time evolution of the participation entropies which quantify the Hilbert space delocalization. We demonstrate that the participation entropies approach, up to a fixed accuracy, their long-time saturation value in times that scale logarithmically with the system size. Exact numerical simulations and tensor network techniques corroborate our findings.
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Haferkamp, Jonas. "Random quantum circuits are approximate unitary t-designs in depth O(nt5+o(1))." Quantum 6 (September 8, 2022): 795. http://dx.doi.org/10.22331/q-2022-09-08-795.
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The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary t-designs. Unitary t-designs are probability distributions that mimic Haar randomness up to tth moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth O(nt10.5) are approximate unitary t-designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by Ω(n−1t−9.5). We improve this lower bound to Ω(n−1t−4−o(1)), where the o(1) term goes to 0 as t→∞. A direct consequence of this scaling is that random quantum circuits generate approximate unitary t-designs in depth O(nt5+o(1)). Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.
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Oszmaniec, Michal, Adam Sawicki, and Michal Horodecki. "Epsilon-Nets, Unitary Designs, and Random Quantum Circuits." IEEE Transactions on Information Theory 68, no. 2 (February 2022): 989–1015. http://dx.doi.org/10.1109/tit.2021.3128110.
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Zhang, Qi, and Guang-Ming Zhang. "Noise-Induced Entanglement Transition in One-Dimensional Random Quantum Circuits." Chinese Physics Letters 39, no. 5 (May 1, 2022): 050302. http://dx.doi.org/10.1088/0256-307x/39/5/050302.
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A random quantum circuit is a minimally structured model to study entanglement dynamics of many-body quantum systems. We consider a one-dimensional quantum circuit with noisy Haar-random unitary gates using density matrix operator and tensor contraction methods. It is shown that the entanglement evolution of the random quantum circuits is properly characterized by the logarithmic entanglement negativity. By performing exact numerical calculations, we find that, as the physical error rate is decreased below a critical value p c ≈ 0.056, the logarithmic entanglement negativity changes from the area law to the volume law, giving rise to an entanglement transition. The critical exponent of the correlation length can be determined from the finite-size scaling analysis, revealing the universal dynamic property of the noisy intermediate-scale quantum devices.
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Bertini, Bruno, Pavel Kos, and Tomaž Prosen. "Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits." Communications in Mathematical Physics 387, no. 1 (July 3, 2021): 597–620. http://dx.doi.org/10.1007/s00220-021-04139-2.
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Hangleiter, Dominik, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert. "Anticoncentration theorems for schemes showing a quantum speedup." Quantum 2 (May 22, 2018): 65. http://dx.doi.org/10.22331/q-2018-05-22-65.
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One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture.
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Cleve, Richard, Debbie Leung, Li Liu, and Chunhao Wang. "Near-linear constructions of exact unitary 2-designs." Quantum Information and Computation 16, no. 9&10 (July 2016): 721–56. http://dx.doi.org/10.26421/qic16.9-10-1.
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A unitary 2-design can be viewed as a quantum analogue of a 2-universal hash function: it is indistinguishable from a truly random unitary by any procedure that queries it twice. We show that exact unitary 2-designs on n qubits can be implemented by quantum circuits consisting of Oe(n) elementary gates in logarithmic depth. This is essentially a quadratic improvement in size (and in width times depth) over all previous implementations that are exact or approximate (for sufficiently strong approximations).
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Brandao, Fernando G. S. L., and Michal Horodecki. "Exponential quantum speed-ups are generic." Quantum Information and Computation 13, no. 11&12 (November 2013): 901–24. http://dx.doi.org/10.26421/qic13.11-12-1.
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A central problem in quantum computation is to understand which quantum circuits are useful for exponential speed-ups over classical computation. We address this question in the setting of query complexity and show that for almost any sufficiently long quantum circuit one can construct a black-box problem which is solved by the circuit with a constant number of quantum queries, but which requires exponentially many classical queries, even if the classical machine has the ability to postselect. We prove the result in two steps. In the first, we show that almost any element of an approximate unitary 3-design is useful to solve a certain black-box problem efficiently. The problem is based on a recent oracle construction of Aaronson and gives an exponential separation between quantum and classical post-selected bounded-error query complexities. In the second step, which may be of independent interest, we prove that linear-sized random quantum circuits give an approximate unitary 3-design. The key ingredient in the proof is a technique from quantum many-body theory to lower bound the spectral gap of local quantum Hamiltonians.
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Farshi, Tom, Daniele Toniolo, Carlos E. González-Guillén, Álvaro M. Alhambra, and Lluis Masanes. "Mixing and localization in random time-periodic quantum circuits of Clifford unitaries." Journal of Mathematical Physics 63, no. 3 (March 1, 2022): 032201. http://dx.doi.org/10.1063/5.0054863.
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How much do local and time-periodic dynamics resemble a random unitary? In the present work, we address this question by using the Clifford formalism from quantum computation. We analyze a Floquet model with disorder, characterized by a family of local, time-periodic, and random quantum circuits in one spatial dimension. We observe that the evolution operator enjoys an extra symmetry at times that are a half-integer multiple of the period. With this, we prove that after the scrambling time, namely, when any initial perturbation has propagated throughout the system, the evolution operator cannot be distinguished from a (Haar) random unitary when all qubits are measured with Pauli operators. This indistinguishability decreases as time goes on, which is in high contrast to the more studied case of (time-dependent) random circuits. We also prove that the evolution of Pauli operators displays a form of mixing. These results require the dimension of the local subsystem to be large. In the opposite regime, our system displays a novel form of localization, produced by the appearance of effective one-sided walls, which prevent perturbations from crossing the wall in one direction but not the other.
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Kretschmer, William. "The Quantum Supremacy Tsirelson Inequality." Quantum 5 (October 7, 2021): 560. http://dx.doi.org/10.22331/q-2021-10-07-560.
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A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit C on n qubits and a sample z∈{0,1}n, the benchmark involves computing |⟨z|C|0n⟩|2, i.e. the probability of measuring z from the output distribution of C on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given C can output a string z such that |⟨z|C|0n⟩|2 is substantially larger than 12n (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit C, sampling z from the output distribution of C achieves |⟨z|C|0n⟩|2≈22n on average (Arute et al., 2019).In analogy with the Tsirelson inequality from quantum nonlocal correlations, we ask: can a polynomial-time quantum algorithm do substantially better than 22n? We study this question in the query (or black box) model, where the quantum algorithm is given oracle access to C. We show that, for any ε≥1poly(n), outputting a sample z such that |⟨z|C|0n⟩|2≥2+ε2n on average requires at least Ω(2n/4poly(n)) queries to C, but not more than O(2n/3) queries to C, if C is either a Haar-random n-qubit unitary, or a canonical state preparation oracle for a Haar-random n-qubit state. We also show that when C samples from the Fourier distribution of a random Boolean function, the naive algorithm that samples from C is the optimal 1-query algorithm for maximizing |⟨z|C|0n⟩|2 on average.
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Bentsen, Gregory, Yingfei Gu, and Andrew Lucas. "Fast scrambling on sparse graphs." Proceedings of the National Academy of Sciences 116, no. 14 (March 21, 2019): 6689–94. http://dx.doi.org/10.1073/pnas.1811033116.
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Given a quantum many-body system with few-body interactions, how rapidly can quantum information be hidden during time evolution? The fast-scrambling conjecture is that the time to thoroughly mix information among N degrees of freedom grows at least logarithmically in N. We derive this inequality for generic quantum systems at infinite temperature, bounding the scrambling time by a finite decay time of local quantum correlations at late times. Using Lieb–Robinson bounds, generalized Sachdev–Ye–Kitaev models, and random unitary circuits, we propose that a logarithmic scrambling time can be achieved in most quantum systems with sparse connectivity. These models also elucidate how quantum chaos is not universally related to scrambling: We construct random few-body circuits with infinite Lyapunov exponent but logarithmic scrambling time. We discuss analogies between quantum models on graphs and quantum black holes and suggest methods to experimentally study scrambling with as many as 100 sparsely connected quantum degrees of freedom.
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Hastings, Matthew B. "Turning gate synthesis errors into incoherent errors." Quantum Information and Computation 17, no. 5&6 (April 2017): 488–94. http://dx.doi.org/10.26421/qic17.5-6-7.
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Using error correcting codes and fault tolerant techniques, it is possible, at least in theory, to produce logical qubits with significantly lower error rates than the underlying physical qubits. Suppose, however, that the gates that act on these logical qubits are only approximation of the desired gate. This can arise, for example, in synthesizing a single qubit unitary from a set of Clifford and T gates; for a generic such unitary, any finite sequence of gates only approximates the desired target. In this case, errors in the gate can add coherently so that, roughly, the error epsilon in the unitary of each gate must scale as epsilon < 1/N, where N is the number of gates. If, however, one has the option of synthesizing one of several unitaries near the desired target, and if an average of these options is closer to the target, we give some elementary bounds showing cases in which the errors can be made to add incoherently by averaging over random choices, so that, roughly, one needs epsilon < 1/ √ N. We remark on one particular application to distilling magic states where this effect happens automatically in the usual circuits.
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Akhtar, Ahmed A., Hong-Ye Hu, and Yi-Zhuang You. "Scalable and Flexible Classical Shadow Tomography with Tensor Networks." Quantum 7 (June 1, 2023): 1026. http://dx.doi.org/10.22331/q-2023-06-01-1026.
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Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
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Elben, Andreas, Jinlong Yu, Guanyu Zhu, Mohammad Hafezi, Frank Pollmann, Peter Zoller, and Benoît Vermersch. "Many-body topological invariants from randomized measurements in synthetic quantum matter." Science Advances 6, no. 15 (April 2020): eaaz3666. http://dx.doi.org/10.1126/sciadv.aaz3666.
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Many-body topological invariants, as quantized highly nonlocal correlators of the many-body wave function, are at the heart of the theoretical description of many-body topological quantum phases, including symmetry-protected and symmetry-enriched topological phases. Here, we propose and analyze a universal toolbox of measurement protocols to reveal many-body topological invariants of phases with global symmetries, which can be implemented in state-of-the-art experiments with synthetic quantum systems, such as Rydberg atoms, trapped ions, and superconducting circuits. The protocol is based on extracting the many-body topological invariants from statistical correlations of randomized measurements, implemented with local random unitary operations followed by site-resolved projective measurements. We illustrate the technique and its application in the context of the complete classification of bosonic symmetry-protected topological phases in one dimension, considering in particular the extended Su-Schrieffer-Heeger spin model, as realized with Rydberg tweezer arrays.
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Mezher, Rawad, Joe Ghalbouni, Joseph Dgheim, and Damian Markham. "On Unitary t-Designs from Relaxed Seeds." Entropy 22, no. 1 (January 12, 2020): 92. http://dx.doi.org/10.3390/e22010092.
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The capacity to randomly pick a unitary across the whole unitary group is a powerful tool across physics and quantum information. A unitary t-design is designed to tackle this challenge in an efficient way, yet constructions to date rely on heavy constraints. In particular, they are composed of ensembles of unitaries which, for technical reasons, must contain inverses and whose entries are algebraic. In this work, we reduce the requirements for generating an ε -approximate unitary t-design. To do so, we first construct a specific n-qubit random quantum circuit composed of a sequence of randomly chosen 2-qubit gates, chosen from a set of unitaries which is approximately universal on U ( 4 ) , yet need not contain unitaries and their inverses nor are in general composed of unitaries whose entries are algebraic; dubbed r e l a x e d seed. We then show that this relaxed seed, when used as a basis for our construction, gives rise to an ε -approximate unitary t-design efficiently, where the depth of our random circuit scales as p o l y ( n , t , l o g ( 1 / ε ) ) , thereby overcoming the two requirements which limited previous constructions. We suspect the result found here is not optimal and can be improved; particularly because the number of gates in the relaxed seeds introduced here grows with n and t. We conjecture that constant sized seeds such as those which are usually present in the literature are sufficient.
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Huang, Xing, and Binchao Zhang. "Growth of a Renormalized Operator as a Probe of Chaos." Advances in High Energy Physics 2022 (October 10, 2022): 1–8. http://dx.doi.org/10.1155/2022/9216427.
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We propose that the size of an operator evolved under holographic renormalization group flow shall grow linearly with the scale and interpret this behavior as a manifestation of the saturation of the chaos bound. To test this conjecture, we study the operator growth in two different toy models. The first one is a MERA-like tensor network built from a random unitary circuit with the operator size defined using the integrated out-of-time-ordered correlator (OTOC). The second model is an error-correcting code of perfect tensors, and the operator size is computed using the number of single-site physical operators that realize the logical operator. In both cases, we observe linear growth.
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Mooney, Gary J., Charles D. Hill, and Lloyd C. L. Hollenberg. "Cost-optimal single-qubit gate synthesis in the Clifford hierarchy." Quantum 5 (February 15, 2021): 396. http://dx.doi.org/10.22331/q-2021-02-15-396.
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For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current procedures do not yet support individual assignment of base gate costs and many do not support extended sets of universal base gates. We analysed cost-optimal sequences using an exhaustive search based on Dijkstra’s pathfinding algorithm for the canonical Clifford+Tset of base gates and compared them to when additionally includingZ-rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced toT-counts by recursively applying aZ-rotation catalyst circuit. Second, costs were assigned as the average numbers of raw (i.e. physical level) magic states required to directly distil and implement the gates fault-tolerantly. We found that the average sequence cost decreases by up to54±3%when using theZ-rotation catalyst circuit approach and by up to33±2%when using the magic state distillation approach. In addition, we investigated observed limitations of certain assignments of base gate costs by developing an analytic model to estimate the proportion of sets ofZ-rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.
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Yuan, Charles, and Michael Carbin. "Tower: data structures in Quantum superposition." Proceedings of the ACM on Programming Languages 6, OOPSLA2 (October 31, 2022): 259–88. http://dx.doi.org/10.1145/3563297.
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Emerging quantum algorithms for problems such as element distinctness, subset sum, and closest pair demonstrate computational advantages by relying on abstract data structures. Practically realizing such an algorithm as a program for a quantum computer requires an efficient implementation of the data structure whose operations correspond to unitary operators that manipulate quantum superpositions of data. To correctly operate in superposition, an implementation must satisfy three properties --- reversibility, history independence, and bounded-time execution. Standard implementations, such as the representation of an abstract set as a hash table, fail these properties, calling for tools to develop specialized implementations. In this work, we present Core Tower, the first language for quantum programming with random-access memory. Core Tower enables the developer to implement data structures as pointer-based, linked data. It features a reversible semantics enabling every valid program to be translated to a unitary quantum circuit. We present Boson, the first memory allocator that supports reversible, history-independent, and constant-time dynamic memory allocation in quantum superposition. We also present Tower, a language for quantum programming with recursively defined data structures. Tower features a type system that bounds all recursion using classical parameters as is necessary for a program to execute on a quantum computer. Using Tower, we implement Ground, the first quantum library of data structures, including lists, stacks, queues, strings, and sets. We provide the first executable implementation of sets that satisfies all three mandated properties of reversibility, history independence, and bounded-time execution.
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Wang, Chunhao, and Leonard Wossnig. "A quantum algorithm for simulating non-sparse Hamiltonians." Quantum Information and Computation 20, no. 7&8 (June 2020): 597–615. http://dx.doi.org/10.26421/qic20.7-8-5.
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We present a quantum algorithm for simulating the dynamics of Hamiltonians that are not necessarily sparse. Our algorithm is based on the input model where the entries of the Hamiltonian are stored in a data structure in a quantum random access memory (qRAM) which allows for the efficient preparation of states that encode the rows of the Hamiltonian. We use a linear combination of quantum walks to achieve poly-logarithmic dependence on precision. The time complexity of our algorithm, measured in terms of the circuit depth, is O(t\sqrt{N}\norm{H}\,\polylog(N, t\norm{H}, 1/\epsilon)), where t is the evolution time, $N$ is the dimension of the system, and $\epsilon$ is the error in the final state, which we call precision. Our algorithm can be directly applied as a subroutine for unitary implementation and quantum linear systems solvers, achieving \widetilde{O}(\sqrt{N}) dependence for both applications.
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Liu, Hong, and Shreya Vardhan. "Void formation in operator growth, entanglement, and unitarity." Journal of High Energy Physics 2021, no. 3 (March 2021). http://dx.doi.org/10.1007/jhep03(2021)159.
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Abstract The structure of the Heisenberg evolution of operators plays a key role in explaining diverse processes in quantum many-body systems. In this paper, we discuss a new universal feature of operator evolution: an operator can develop a void during its evolution, where its nontrivial parts become separated by a region of identity operators. Such processes are present in both integrable and chaotic systems, and are required by unitarity. We show that void formation has important implications for unitarity of entanglement growth and generation of mutual information and multipartite entanglement. We study explicitly the probability distributions of void formation in a number of unitary circuit models, and conjecture that in a quantum chaotic system the distribution is given by the one we find in random unitary circuits, which we refer to as the random void distribution. We also show that random unitary circuits lead to the same pattern of entanglement growth for multiple intervals as in (1 + 1)-dimensional holographic CFTs after a global quench, which can be used to argue that the random void distribution leads to maximal entanglement growth.
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Kasim, Yusuf, and Tomaz Prosen. "Dual unitary circuits in random geometries." Journal of Physics A: Mathematical and Theoretical, January 10, 2023. http://dx.doi.org/10.1088/1751-8121/acb1e0.
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Abstract Recently introduced dual unitary brickwork circuits have been recognised as paradigmatic exactly solvable quantum chaotic many-body systems with tunable degree of ergodicity and mixing. Here we show that regularity of the circuit lattice is not crucial for exact solvability. We consider a circuit where random 2-qubit dual unitary gates sit at intersections of random arrangements of straight lines in two dimensions (mikado) and analytically compute the variance of the spatio-temporal correlation function of local operators. Note that the average correlator vanishes due to local Haar randomness of the gates. The result can be physically motivated for two random mikado settings. The first corresponds to the thermal state of free particles carrying internal qubit degrees of freedom which experience interaction at kinematic crossings, while the second represents rotationally symmetric (random euclidean) space-time.
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Nahum, Adam, Sagar Vijay, and Jeongwan Haah. "Operator Spreading in Random Unitary Circuits." Physical Review X 8, no. 2 (April 11, 2018). http://dx.doi.org/10.1103/physrevx.8.021014.
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Haferkamp, Jonas, Philippe Faist, Naga B. T. Kothakonda, Jens Eisert, and Nicole Yunger Halpern. "Linear growth of quantum circuit complexity." Nature Physics, March 28, 2022. http://dx.doi.org/10.1038/s41567-022-01539-6.
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AbstractThe complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.
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Perlin, Michael A., Zain H. Saleem, Martin Suchara, and James C. Osborn. "Quantum circuit cutting with maximum-likelihood tomography." npj Quantum Information 7, no. 1 (April 23, 2021). http://dx.doi.org/10.1038/s41534-021-00390-6.
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AbstractWe introduce maximum-likelihood fragment tomography (MLFT) as an improved circuit cutting technique for running clustered quantum circuits on quantum devices with a limited number of qubits. In addition to minimizing the classical computing overhead of circuit cutting methods, MLFT finds the most likely probability distribution for the output of a quantum circuit, given the measurement data obtained from the circuit’s fragments. We demonstrate the benefits of MLFT for accurately estimating the output of a fragmented quantum circuit with numerical experiments on random unitary circuits. Finally, we show that circuit cutting can estimate the output of a clustered circuit with higher fidelity than full circuit execution, thereby motivating the use of circuit cutting as a standard tool for running clustered circuits on quantum hardware.
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Bertini, Bruno, and Lorenzo Piroli. "Scrambling in random unitary circuits: Exact results." Physical Review B 102, no. 6 (August 10, 2020). http://dx.doi.org/10.1103/physrevb.102.064305.
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Oh, Sangchul, and Sabre Kais. "Cutoff phenomenon and entropic uncertainty for random quantum circuits." Electronic Structure, August 22, 2023. http://dx.doi.org/10.1088/2516-1075/acf2d3.
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Abstract How fast a state of a system converges to a stationary state is one of the fundamental questions in science. Some Markov chains and random walks on finite groups are known to exhibit the non-asymptotic convergence to a stationary distribution, called the cutoff phenomenon. Here, we examine how quickly a random quantum circuit could transform a quantum state to a Haar-measure random quantum state. We find that random quantum states, as stationary states of random walks on a unitary group, are invariant under the quantum Fourier transform. Thus the entropic uncertainty of random quantum states has balanced Shannon entropies for the computational bases and the quantum Fourier transform bases. By calculating the Shannon entropy for random quantum states and the Wasserstein distances for the eigenvalues of random quantum circuits, we show that the cutoff phenomenon occurs for the random quantum circuit. It is also demonstrated that the Dyson-Brownian motion for the eigenvalues of a random unitary matrix as a continuous random walk exhibits the cutoff phenomenon. The results here imply that random quantum states could be generated with shallow random circuits.
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Fisher, Matthew P. A., Vedika Khemani, Adam Nahum, and Sagar Vijay. "Random Quantum Circuits." Annual Review of Condensed Matter Physics 14, no. 1 (December 12, 2022). http://dx.doi.org/10.1146/annurev-conmatphys-031720-030658.
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Quantum circuits—built from local unitary gates and local measurements—are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far from equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as dynamical phase transitions in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics. Expected final online publication date for the Annual Review of Condensed Matter Physics, Volume 14 is March 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
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Yu, Sunkyu, and Namkyoo Park. "Heavy tails and pruning in programmable photonic circuits for universal unitaries." Nature Communications 14, no. 1 (April 3, 2023). http://dx.doi.org/10.1038/s41467-023-37611-9.
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AbstractDeveloping hardware for high-dimensional unitary operators plays a vital role in implementing quantum computations and deep learning accelerations. Programmable photonic circuits are singularly promising candidates for universal unitaries owing to intrinsic unitarity, ultrafast tunability and energy efficiency of photonic platforms. Nonetheless, when the scale of a photonic circuit increases, the effects of noise on the fidelity of quantum operators and deep learning weight matrices become more severe. Here we demonstrate a nontrivial stochastic nature of large-scale programmable photonic circuits—heavy-tailed distributions of rotation operators—that enables the development of high-fidelity universal unitaries through designed pruning of superfluous rotations. The power law and the Pareto principle for the conventional architecture of programmable photonic circuits are revealed with the presence of hub phase shifters, allowing for the application of network pruning to the design of photonic hardware. For the Clements design of programmable photonic circuits, we extract a universal architecture for pruning random unitary matrices and prove that “the bad is sometimes better to be removed” to achieve high fidelity and energy efficiency. This result lowers the hurdle for high fidelity in large-scale quantum computing and photonic deep learning accelerators.
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Bulchandani, Vir B., S. L. Sondhi, and J. T. Chalker. "Random-Matrix Models of Monitored Quantum Circuits." Journal of Statistical Physics 191, no. 5 (May 3, 2024). http://dx.doi.org/10.1007/s10955-024-03273-0.
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AbstractWe study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter–Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker–Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally.
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Bera, Anindita, and Sudipto Singha Roy. "Growth of genuine multipartite entanglement in random unitary circuits." Physical Review A 102, no. 6 (December 30, 2020). http://dx.doi.org/10.1103/physreva.102.062431.
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Zhou, Tianci, and Adam Nahum. "Emergent statistical mechanics of entanglement in random unitary circuits." Physical Review B 99, no. 17 (May 20, 2019). http://dx.doi.org/10.1103/physrevb.99.174205.
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Arnaud, Ludovic, and Daniel Braun. "Efficiency of producing random unitary matrices with quantum circuits." Physical Review A 78, no. 6 (December 17, 2008). http://dx.doi.org/10.1103/physreva.78.062329.
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Riddell, Jonathon, Curt von Keyserlingk, Tomaž Prosen, and Bruno Bertini. "Structural stability hypothesis of dual unitary quantum chaos." Physical Review Research 6, no. 3 (August 29, 2024). http://dx.doi.org/10.1103/physrevresearch.6.033226.
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Having spectral correlations that, over small enough energy scales, are described by random matrix theory is regarded as the most general defining feature of quantum chaotic systems as it applies in the many-body setting and away from any semiclassical limit. Although this property is extremely difficult to prove analytically for generic many-body systems, a rigorous proof has been achieved for dual-unitary circuits—a special class of local quantum circuits that remain unitary upon swapping space and time. Here we consider the fate of this property when moving from dual-unitary to generic quantum circuits focusing on the , i.e., the Fourier transform of the two-point correlation. We begin with a numerical survey that, in agreement with previous studies, suggests that there exists a finite region in parameter space where dual-unitary physics is stable and spectral correlations are still described by random matrix theory, although up to a maximal quasienergy scale. To explain these findings, we develop a perturbative expansion: it recovers the random matrix theory predictions, provided the terms occurring in perturbation theory obey a relatively simple set of assumptions. We then provide numerical evidence and a heuristic analytical argument supporting these assumptions. Published by the American Physical Society 2024
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Harrow, Aram W., and Saeed Mehraban. "Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates." Communications in Mathematical Physics, May 4, 2023. http://dx.doi.org/10.1007/s00220-023-04675-z.
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AbstractWe prove that $${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ poly ( t ) · n 1 / D -depth local random quantum circuits with two qudit nearest-neighbor gates on a D-dimensional lattice with n qudits are approximate t-designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was $${{\,\textrm{poly}\,}}(t)\cdot n$$ poly ( t ) · n due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for $$D=1$$ D = 1 . We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($${{\,\mathrm{\textsf{PH}}\,}}$$ PH ) is infinite and that certain counting problems are $$\#{\textsf{P}}$$ # P -hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that $$O(\sqrt{n})$$ O ( n ) depth suffices for anti-concentration. The proof is based on a previous construction of t-designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size $$O(n\ln ^2 n)$$ O ( n ln 2 n ) corresponding to depth $$O(\ln ^3 n)$$ O ( ln 3 n ) . We also show a lower bound of $$\Omega (n \ln n)$$ Ω ( n ln n ) for the size of such circuit in this case. We also prove that anti-concentration is possible in depth $$O(\ln n \ln \ln n)$$ O ( ln n ln ln n ) (size $$O(n \ln n \ln \ln n)$$ O ( n ln n ln ln n ) ) using a different model.
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Kalsi, Tara, Alessandro Romito, and Henning Schomerus. "Three-fold way of entanglement dynamics in monitored quantum circuits." Journal of Physics A: Mathematical and Theoretical, May 20, 2022. http://dx.doi.org/10.1088/1751-8121/ac71e8.
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Abstract We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson's three circular ensembles (circular unitary, orthogonal, and symplectic ensembles; CUE, COE and CSE). We utilise the established model of a one-dimensional circuit evolving under alternating local random unitary gates and projective measurements performed with tunable rate, which for gates drawn from the CUE is known to display a transition from extensive to intensive entanglement scaling as the measurement rate is increased. By contrasting this case to the COE and CSE, we obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements. For this, we combine exact analytical random-matrix results for the entanglement generated by the individual gates in the different ensembles, and numerical results for the complete quantum circuit. These considerations include an efficient rephrasing of the statistical entangling power in terms of a characteristic entanglement matrix capturing the essence of Cartan's KAK decomposition, and a general result for the eigenvalue statistics of antisymmetric matrices associated with the CSE.
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Fan, Ruihua, Sagar Vijay, Ashvin Vishwanath, and Yi-Zhuang You. "Self-organized error correction in random unitary circuits with measurement." Physical Review B 103, no. 17 (May 27, 2021). http://dx.doi.org/10.1103/physrevb.103.174309.
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Bao, Yimu, Soonwon Choi, and Ehud Altman. "Theory of the phase transition in random unitary circuits with measurements." Physical Review B 101, no. 10 (March 3, 2020). http://dx.doi.org/10.1103/physrevb.101.104301.
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Sierant, Piotr, Marco Schirò, Maciej Lewenstein, and Xhek Turkeshi. "Entanglement Growth and Minimal Membranes in ( d+1 ) Random Unitary Circuits." Physical Review Letters 131, no. 23 (December 8, 2023). http://dx.doi.org/10.1103/physrevlett.131.230403.
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Yao, Jiangtian, and Pieter W. Claeys. "Temporal entanglement barriers in dual-unitary Clifford circuits with measurements." Physical Review Research 6, no. 4 (October 29, 2024). http://dx.doi.org/10.1103/physrevresearch.6.043077.
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We study temporal entanglement in dual-unitary Clifford circuits with probabilistic measurements preserving spatial unitarity. We exactly characterize the temporal entanglement barrier in the measurement-free regime, exhibiting ballistic growth and decay and a volume-law peak. In the presence of measurements, we relate the temporal entanglement to the scrambling properties of the circuit. For “good scramblers” measurements do not qualitatively change the temporal entanglement profile but only result in a reduced entanglement velocity, whereas for “poor scramblers” the initial ballistic growth of temporal entanglement with bath size is modified to diffusive. This difference is understood through a mapping of the underlying operator dynamics to a biased and an unbiased persistent random walk, respectively. In the latter case measurements additionally modify the ballistic decay to the perfect dephaser limit, with vanishing temporal entanglement, to an exponential decay, which we describe through a spatial transfer matrix method. This spatial dynamics is shown to be described by a non-Hermitian hopping model, exhibiting a PT-breaking transition at a critical measurement rate p=1/2. In all cases the peak value of the temporal entanglement barrier exhibits volume-law scaling for all measurement rates. Published by the American Physical Society 2024
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Smith, Kevin C., Abid Khan, Bryan K. Clark, S. M. Girvin, and Tzu-Chieh Wei. "Constant-Depth Preparation of Matrix Product States with Adaptive Quantum Circuits." PRX Quantum 5, no. 3 (September 4, 2024). http://dx.doi.org/10.1103/prxquantum.5.030344.
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Adaptive quantum circuits, which combine local unitary gates, midcircuit measurements, and feedforward operations, have recently emerged as a promising avenue for efficient state preparation, particularly on near-term quantum devices limited to shallow-depth circuits. Matrix product states (MPS) comprise a significant class of many-body entangled states, efficiently describing the ground states of one-dimensional gapped local Hamiltonians and finding applications in a number of recent quantum algorithms. Recently, it has been shown that the Affleck-Kennedy-Lieb-Tasaki state—a paradigmatic example of an MPS—can be exactly prepared with an adaptive quantum circuit of constant depth, an impossible feat with local unitary gates alone due to its nonzero correlation length [Smith , PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this approach and demonstrate that a diverse class of MPS can be exactly prepared using constant-depth adaptive quantum circuits, outperforming theoretically optimal preparation with unitary circuits. We show that this class includes short- and long-ranged entangled MPS, symmetry-protected topological (SPT) and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families of states with tunable correlation length. Moreover, we illustrate the utility of our framework for designing constant-depth sampling protocols, such as for random MPS or for generating MPS in a particular SPT phase. We present sufficient conditions for particular MPS to be preparable in constant time, with global on-site symmetry playing a pivotal role. Altogether, this work demonstrates the immense promise of adaptive quantum circuits for efficiently preparing many-body entangled states and provides explicit algorithms that outperform known protocols to prepare an essential class of states. Published by the American Physical Society 2024
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Haferkamp, J., F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth. "Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates." Communications in Mathematical Physics, November 12, 2022. http://dx.doi.org/10.1007/s00220-022-04507-6.
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AbstractMany quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject $$O(t^{4}\log ^{2}(t)\log (1/\varepsilon ))$$ O ( t 4 log 2 ( t ) log ( 1 / ε ) ) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an $$\varepsilon $$ ε -approximate t-design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.
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Lovas, Izabella, Utkarsh Agrawal, and Sagar Vijay. "Quantum Coding Transitions in the Presence of Boundary Dissipation." PRX Quantum 5, no. 3 (August 7, 2024). http://dx.doi.org/10.1103/prxquantum.5.030327.
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We investigate phase transitions in the encoding of quantum information in a quantum many-body system due to the competing effects of unitary scrambling and boundary dissipation. Specifically, we study the fate of quantum information in a one-dimensional qudit chain, subject to local unitary quantum circuit evolution in the presence of depolarizing noise at the boundary. If the qudit chain initially contains a finite amount of locally accessible quantum information, unitary evolution in the presence of boundary dissipation allows this information to remain partially protected when the dissipation is sufficiently weak, and up to timescales growing linearly in the system size L. In contrast, for strong enough dissipation, this information is completely lost to the dissipative environment. We analytically investigate this “quantum coding transition” by considering dynamics involving Haar-random, local unitary gates, and confirm our predictions in numerical simulations of Clifford quantum circuits. Scrambling the quantum information in the qudit chain with a unitary circuit of depth O(logL) before the onset of dissipation can perfectly protect the information until late times. The nature of the coding transition changes when the dynamics extend for times much longer than L. We further show that at weak dissipation, it is possible to code at a finite rate, i.e., a fraction of the many-body Hilbert space of the qudit chain can be used to encode quantum information. Published by the American Physical Society 2024
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Wiersema, Roeland, Cunlu Zhou, Juan Felipe Carrasquilla, and Yong Baek Kim. "Measurement-induced entanglement phase transitions in variational quantum circuits." SciPost Physics 14, no. 6 (June 8, 2023). http://dx.doi.org/10.21468/scipostphys.14.6.147.
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Variational quantum algorithms (VQAs), which classically optimize a parametrized quantum circuit to solve a computational task, promise to advance our understanding of quantum many-body systems and improve machine learning algorithms using near-term quantum computers. Prominent challenges associated with this family of quantum-classical hybrid algorithms are the control of quantum entanglement and quantum gradients linked to their classical optimization. Known as the barren plateau phenomenon, these quantum gradients may rapidly vanish in the presence of volume-law entanglement growth, which poses a serious obstacle to the practical utility of VQAs. Inspired by recent studies of measurement-induced entanglement transition in random circuits, we investigate the entanglement transition in variational quantum circuits endowed with intermediate projective measurements. Considering the Hamiltonian Variational Ansatz (HVA) for the XXZ model and the Hardware Efficient Ansatz (HEA), we observe a measurement-induced entanglement transition from volume-law to area-law with increasing measurement rate. Moreover, we provide evidence that the transition belongs to the same universality class of random unitary circuits. Importantly, the transition coincides with a “landscape transition” from severe to mild/no barren plateaus in the classical optimization. Our work may provide an avenue for improving the trainability of quantum circuits by incorporating intermediate measurement protocols in currently available quantum hardware.
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Jiang, Weiwen, Jinjun Xiong, and Yiyu Shi. "A co-design framework of neural networks and quantum circuits towards quantum advantage." Nature Communications 12, no. 1 (January 25, 2021). http://dx.doi.org/10.1038/s41467-020-20729-5.
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AbstractDespite the pursuit of quantum advantages in various applications, the power of quantum computers in executing neural network has mostly remained unknown, primarily due to a missing tool that effectively designs a neural network suitable for quantum circuit. Here, we present a neural network and quantum circuit co-design framework, namely QuantumFlow, to address the issue. In QuantumFlow, we represent data as unitary matrices to exploit quantum power by encoding n = 2k inputs into k qubits and representing data as random variables to seamlessly connect layers without measurement. Coupled with a novel algorithm, the cost complexity of the unitary matrices-based neural computation can be reduced from O(n) in classical computing to O(polylog(n)) in quantum computing. Results show that on MNIST dataset, QuantumFlow can achieve an accuracy of 94.09% with a cost reduction of 10.85 × against the classical computer. All these results demonstrate the potential for QuantumFlow to achieve the quantum advantage.
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Peetz, Joseph, Scott E. Smart, Spyros Tserkis, and Prineha Narang. "Simulation of open quantum systems via low-depth convex unitary evolutions." Physical Review Research 6, no. 2 (June 10, 2024). http://dx.doi.org/10.1103/physrevresearch.6.023263.
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Simulating physical systems on quantum devices is one of the most promising applications of quantum technology. Current quantum approaches to simulating open quantum systems are still practically challenging on NISQ-era devices, because they typically require ancilla qubits and extensive controlled sequences. In this work, we propose a hybrid quantum-classical approach for simulating a class of open system dynamics called random-unitary channels. These channels naturally decompose into a series of convex unitary evolutions, which can then be efficiently sampled and run as independent circuits. The method does not require deep ancilla frameworks and thus can be implemented with lower noise costs. We implement simulations of open quantum systems up to dozens of qubits and with large channel ranks. Published by the American Physical Society 2024
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de Queiroz, S. L. A. "Rare-event properties in a classical stochastic model describing the evolution of random unitary circuits." Physical Review E 104, no. 3 (September 16, 2021). http://dx.doi.org/10.1103/physreve.104.034122.
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Garcia-Escartin, Juan Carlos. "Finding eigenvectors with a quantum variational algorithm." Quantum Information Processing 23, no. 7 (June 25, 2024). http://dx.doi.org/10.1007/s11128-024-04461-3.
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AbstractThis paper presents a hybrid variational quantum algorithm that finds a random eigenvector of a unitary matrix with a known quantum circuit. The algorithm is based on the SWAP test on trial states generated by a parametrized quantum circuit. The eigenvector is described by a compact set of classical parameters that can be used to reproduce the found approximation to the eigenstate on demand. This variational eigenvector finder can be adapted to solve the generalized eigenvalue problem, to find the eigenvectors of normal matrices and to perform quantum principal component analysis on unknown input mixed states. These algorithms can all be run with low-depth quantum circuits, suitable for an efficient implementation on noisy intermediate-scale quantum computers and, with some restrictions, on linear optical systems. In full-scale quantum computers, where there might be optimization problems due to barren plateaus in larger systems, the proposed algorithms can be used as a primitive to boost known quantum algorithms. Limitations and potential applications are discussed.