Quantum complexity phase transitions in monitored random circuits

A Busy March Meeting Is Brewing in Seattle

Quantum computing is a multidisciplinary field at the intersectionof quantum physics, computer science, and mathematics, aiming to har-ness quantum mechanical phenomena for computational advantage. Thisarticle provides a comprehensive review of both the foundational princi-ples and advanced theoretical models of quantum computing. We beginwith the basic framework of quantum computation, introducing the con-cept of qubits as two-level quantum systems, the linear algebra of Hilbertspaces, Dirac’s bra–ket notation, and the postulates of quantum mechan-ics (state superposition, unitary evolution, and projective measurement).We then discuss quantum logic gates and circuits, highlighting how clas-sical reversible computation principles extend to universal quantum gatesets. Next, we survey key quantum algorithms (from Shor’s factoring toGrover’s searching and beyond) and the complexity-theoretic implicationsof quantum computers, contrasting the class BQP with classical complex-ity classes. We cover the theoretical foundations of quantum error cor-rection and fault tolerance, explaining how quantum information can beprotected from decoherence using redundancy and how fault-tolerant pro-tocols can, in principle, allow scalable quantum computation despite noisyhardware. We then explore advanced models and paradigms of quantumcomputation, including measurement-based quantum computing (one-wayquantum computing), topological quantum computation with anyons, adi-abatic quantum computing and quantum annealing, continuous-variablequantum computing, and related approaches. We also provide an overviewof quantum information theory as it relates to computation, discussingentanglement, quantum entropy, and information-theoretic limits like theHolevo bound. Finally, we review the leading physical implementations(quantum hardware models) of quantum computers—trapped ions, super-conducting circuits, photonic systems, solid-state spin qubits, and oth-ers—outlining the challenges and achievements of each approach.

Quantum computers are emerging technologies expected to become important tools for exploring various aspects of fundamental physics in the future. Therefore, we pose the question of whether quantum computers can help us to study the Page curve and the black hole information dynamics, which has been a key focus in fundamental physics. In this regard, we rigorously examine the qubit transport model, a toy qubit model of black hole evaporation on IBM’s superconducting quantum computers, to shed light on this question. Specifically, we implement the quantum simulation of the scrambling dynamics in black holes using an efficient random unitary circuit. Furthermore, we employ the swap-based many-body interference protocol and the randomized measurement protocol to measure the entanglement entropy of Hawking radiation qubits in this model. Finally, by incorporating quantum error mitigation techniques into our challenging implementation of entanglement entropy measurement protocols on the IBM quantum hardware, we accurately determine the Rényi entropy in the qubit transport model, thus showcasing the utility of quantum computers for future investigations of complex quantum systems.

We analyze the dynamics of entanglement entropy in a generic quantum many-body open system from the perspective of quantum information and error corrections. We introduce a random unitary circuit model with intermittent projective measurements, in which the degree of information scrambling by the unitary and the rate of projective measurements are independently controlled. This model displays two stable phases, characterized by the volume-law and area-law scaling entanglement entropy in steady states. The transition between the two phases is understood from the point of view of quantum error correction: the chaotic unitary evolution protects quantum information from projective measurements that act as errors. A phase transition occurs when the rate of errors exceeds a threshold that depends on the degree of information scrambling. We confirm these results using numerical simulations and obtain the phase diagram of our model. Our work shows that information scrambling plays a crucial role in understanding the dynamics of entanglement in an open quantum system and relates the entanglement phase transition to changes in quantum channel capacity.

We show that counting the number of collisions (re-sampled bitstrings) when measuring a random quantum circuit provides a practical benchmark for the quality of a quantum computer and a quantitative noise characterization method. We analytically estimate the difference in the expected number of collisions when sampling bitstrings from a pure random state and when sampling from the classical uniform distribution. We show that this quantity, if properly normalized, can be used as a collision anomaly benchmark or as a collision volume test, which is similar to the well-known quantum volume test, with advantages (no classical computing cost) and disadvantages (high sampling cost). We also propose to count the number of cross-collisions between two independent quantum computers running the same random circuit in order to obtain a cross-validation test of the two devices. Finally, we quantify the sampling cost of quantum collision experiments. We find that the sampling cost for running a collision volume test on state-of-the-art processors (e.g., 20 effective clean qubits) is quite small: less than 105 shots. For large-scale experiments in the quantum supremacy regime, the required number of shots for observing a quantum signal in the observed number of collisions is currently infeasible (>1012), but not completely out of reach for near-future technology.

Understanding out-of-equilibrium many-body quantum systems is an essential task in contemporary physics. While advanced numerical methods have been developed, capturing the universal dynamics of generic many-body quantum systems still remains a significant challenge. In this study, we focus on the multichannel Kondo impurity (MCKI) model, an intriguing theoretical model hosting an overscreened Kondo state with non-Fermi-liquid characteristics, which serves as a theoretical platform for investigating universal properties in many-body quantum dynamics. Utilizing the large-$N$ Schwinger-Keldysh approach, we systematically investigate both transient dynamics and long-time quasiequilibrium properties in the MCKI model following a sudden change of the Kondo coupling. Our investigations encompass two distinct initial states: the overscreened Kondo state and the high-temperature Fermi-liquid state. For the overscreened Kondo state initial condition, we observe oscillations in various physical observables, including spin-spin correlations and the Kondo order parameter. These oscillations signify the quantum revival of the entangled state that characterizes the overscreened Kondo state. On the other hand, in the case of the high-temperature Fermi-liquid initial state, the absence of oscillations can be attributed to the dephasing mechanism. Furthermore, we discover that the system reaches a quasiequilibrium on an $O(1)$ timescale. This quasiequilibrium manifests as incoherent thermalization between the impurity and the conduction electrons, in which we observe the nonvanishing effective temperature difference between the Abrikosov fermion representing the impurity spin and the composite boson formed by the Abrikosov fermion and the conduction electrons at the impurity site. This quasiequilibrium can be interpreted as prethermalization. Incorporating the $1/N$ correction, we demonstrate that the system attains complete thermalization on an $O(N)$ timescale. Additionally, we discuss the quantum cooling effect and quantum Boltzmann equations. Our comprehensive study establishes a foundation for investigating quantum many-body systems using large-$N$ quantum field-theory treatment, while our findings reveal several universal properties of quantum dynamics and provide a different perspective on prethermalization.

Recently it has become clear, that a key issue in quantum computation is understanding how interaction with the environment, or decoherence, affects the computational power of quantum computers. We adopt the standard physical method of describing systems which are interwound with their environment by density matrices, and within this framework define a model of decoherence in quantum computation. Our results show that the computational power of decohered quantum computers depends strongly on the amount of parallelism in the computation. We first present a simulation of decohered sequential quantum computers, on a classical probabilistic Turing machine, and prove that the expected slowdown of this simulation is polynomial in time and space of the quantum computation, for any non zero decoherence rate. Similar results hold for quantum computers that are allowed to operate on logarithmic number of qubits at a time. For decohered quantum circuits (with local gates), the situation is more subtle and depends on the decoherence rate, /spl eta/. We find that our simulation is efficient for circuits with decoherence rate /spl eta/ higher than some constant /spl eta//sub 1/ but exponential for a general (random) circuit subjected to decoherence rate lower than some constant /spl eta//sub 2/. The transition from exponential cost to polynomial cost happens in a short range of decoherence rates. We use computer experiments to exhibit the phase transitions in various quantum circuits.

Quantum field lens coding and classification algorithm to predict measurement outcomes

In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison with their symmetric versions (SIC-POVMs). We establish a correspondence between the standard quantum-mechanical formalism and the MIC-POVM-based probability formalism. Within the latter approach, we derive equations for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator. We apply the MIC-POVM-based probability representation to the digital quantum computing model. In particular, for the case of spin-1/2 evolution, we demonstrate identifying a transition of a dissipative quantum dynamics to a completely classical-like stochastic dynamics. One of the most important findings is that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach. Our results give a physical interpretation of quantum computations and pave a way for exploring the resources of noisy intermediate-scale quantum devices.

Interacting many-body quantum systems and their dynamics, while fundamental to modern science and technology, are formidable to simulate and understand. However, by discovering their symmetries, conservation laws, and integrability, one can unravel their intricacies. Here, using up to 124 qubits of a fully programmable quantum computer, we uncover local conservation laws and integrability in one- and two-dimensional periodically-driven spin lattices in a regime previously inaccessible to such detailed analysis. We focus on the paradigmatic example of disorder-induced ergodicity breaking, where we first benchmark the system crossover into a localized regime through anomalies in the one-particle-density-matrix spectrum and other hallmark signatures. We then demonstrate that this regime stems from hidden local integrals of motion by faithfully reconstructing their quantum operators, thus providing a more detailed portrait of the system’s integrable dynamics. Our results demonstrate a versatile strategy for extracting the hidden dynamical structure from noisy experiments on large-scale quantum computers.

Quantum mechanics is the most ground-breaking and fascinating theoretical concept developed in physics during the past century. Much of our present understanding of the microscopic world and its extension into the macroscopic world, including modern technical applications, is based upon quantum mechanics. We have experienced a remarkable development of information and communication technology during the past two decades, to a large extent depending upon successful fabrication of smaller and smaller components and circuits. However, we are finally approaching the physical limits of component miniaturization as we enter a microscopic world ruled by quantum mechanics.

Post-quantum cryptography aims to find quantum-resistant public-key cryptographic schemes, that is, schemes which remain secure against both quantum and standard computers. In fact in 1994, Shor published an algorithm [85] breaking the nowadays used public-key cryptoschemes in polynomial time on a quantum computer. The security of public-key algorithms is based on computationally hard mathematical problems. Among the post-quantum proposals there is the Multivariate Problem: Solving a multivariate polynomial system over a finite field. The general and the most computationally efficient method for solving such systems is computing a Groöbner basis of the ideal they generate. In 2021, Beullens [22] used a strategy involving the MinRank Problem for breaching the most promising multivariate scheme at the time. All the details regarding public-key schemes and in particular the multivariate one can be found within the Introduction. The thesis is divided into three blocks. The goal of Chapter 1 is twofold. It is a collection of algebraic concepts and classical facts that are used through these pages, in particular, Section 1.1 serves this scope. In addition, the problem of solving polynomial systems via Gr¨bner basis is fully explained in Section 1.2. In Chapter 2 and Chapter 3, I present the two main works of my PhD, during which I mainly focused on the analysis of multivariate cryptoschemes twofolds. Firstly, together with my advisor Elisa Gorla, we propose a new definition of ‘random system’ using the concept of generality mutued from algebraic geometry. The details of this work are given in Chapter 2. On the other hand, with the help of Daniel Cabarcas (Universidad Nacional de Colombia), we investigated the SupportMinors Modeling, which is a method for modeling the MinRank Problem used by Beullens in his attack. The specifics of this work can be found in Chapter 3.

Synthesis of multi-qudit hybrid and d-valued quantum logic circuits by decomposition

Motion planning remains a fundamental problem in robotics. Sampling-based algorithms use randomization to allow efficient solutions to this complex problem. As mobile robots and autonomous vehicles become more prevalent in everyday life, motion planning must be applied to increasingly challenging scenarios. Safety has become a paramount concern in motion planning for ensuring robotic applications enrich human lives. To date, many motion planning techniques to increase safety in the face of uncertain and dynamic environments have been developed. This dissertation first addresses distributional safety of Rapidly-Exploring Random Trees (RRT) through our algorithm W-Safe RRT. To acknowledge distributional uncertainty and poor modeling, W-Safe RRT uses the Wasserstein metric to provide a probabilistic bound on the distributional distance between a robot and obstacles. Human-interpretable environmental agent classification allows online safety margin adaptation. We propose and analyze an integrating region method for online classification that increases actor labeling accuracy based on behavioral feature values when compared to state of the art methods. The method performs class assignments based on local maximum likelihood in a created behavioral feature-space, allowing a notion of classification uncertainty. Model-based methods with safety guarantees can quickly become computationally in tractable, especially with multiple agents, higher dimensions, and plentiful unknowns. Sampling based algorithms have been parallelized for computation with multi-core computers and GPUs. We consider the use of quantum algorithms and computers for sampling-based motion planning for the first time. Quantum computing performs operations on superpositions of states and can solve certain problems much more efficiently than classical computers, but introduces previously unseen challenges. With Quantum-RRT, we recast the motion planning problem into a database-search structure and use Quantum Amplitude Amplification to find reachable states in the database with a quadratic performance increase over classical methods. We address two error sources with this method: quantum measurement and quantum oracle errors. We then extend this method to Parallel Quantum-RRT, which uses a manager-worker architecture with multiple parallel quantum workers to increase database search efficiency. We compare algorithm architectures and characterize probabilities of multiple workers finding solutions. Lastly, we test in simulation the quantum algorithms against classical versions in a wide variety of scenarios, concluding that a similar parallelization improvement is to be found in the quantum case as was found in the parallelization of classical RRT.

Modeling many-body quantum systems is widely regarded as one of the most promising applications for near-term noisy quantum computers. However, in the near term, system size limitation will remain a severe barrier for applications in materials science or strongly correlated systems. A promising avenue of research is to combine many-body physics with machine learning for the classification of distinct phases. We present a workflow that synergizes quantum computing, many-body theory, and quantum machine learning (QML) for studying strongly correlated systems. In particular, it can capture a putative quantum phase transition of the stereotypical strongly correlated system, the Hubbard model. Following the recent proposal of the hybrid quantum-classical algorithm for the two-site dynamical mean-field theory (DMFT), we present a modification that allows the self-consistent solution of the single bath site DMFT. The modified algorithm can be generalized for multiple bath sites. This approach is used to generate a database of zero-temperature wavefunctions of the Hubbard model within the DMFT approximation. We then use a QML algorithm to distinguish between the metallic phase and the Mott insulator phase to capture the metal-to-Mott insulator phase transition. We train a recently proposed quantum convolutional neural network (QCNN) and then utilize the QCNN as a quantum classifier to capture the phase transition region. This work provides a recipe for application to other phase transitions in strongly correlated systems and represents an exciting application of small-scale quantum devices realizable with near-term technology.