In a recent milestone experiment, Google's processor Sycamore heralded the era of "quantum supremacy" by sampling from the output of (pseudo-)random circuits. We show that such random circuits provide tailor-made building blocks for simulating quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. Specifically, we propose an algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution to study hydrodynamics and to extract transport coefficients in the linear response regime. We numerically demonstrate the algorithm by simulating the buildup of spatiotemporal correlation functions in one- and two-dimensional quantum spin systems, where we particularly scrutinize the inevitable impact of errors present in any realistic implementation. Importantly, we find that the hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, which opens the door to reach nontrivial time scales with a small number of gates. While errors within the random circuit are shown to be irrelevant, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware. Our work emphasizes the practical relevance of random circuits on NISQ devices beyond the abstract sampling task.
In a recent milestone experiment, Google's processor Sycamore heralded the era of "quantum supremacy" by sampling from the output of (pseudo-)random circuits. We show that such random circuits provide tailor-made building blocks for simulating quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. Specifically, we propose an algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution to study hydrodynamics and to extract transport coefficients in the linear response regime. We numerically demonstrate the algorithm by simulating the buildup of spatiotemporal correlation functions in one- and two-dimensional quantum spin systems, where we particularly scrutinize the inevitable impact of errors present in any realistic implementation. Importantly, we find that the hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, which opens the door to reach nontrivial time scales with a small number of gates. While errors within the random circuit are shown to be irrelevant, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware. Our work emphasizes the practical relevance of random circuits on NISQ devices beyond the abstract sampling task.
In a recent milestone experiment, Google's processor Sycamore heralded the era of supremacy by sampling from the output of (pseudo-)random circuits. We show that such random circuits provide tailor-made building blocks for simulating quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. Specifically, we propose an algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution to study hydrodynamics and to extract transport coefficients in the linear response regime. We numerically demonstrate the algorithm by simulating the buildup of spatiotemporal correlation functions in one- and two-dimensional quantum spin systems, where we particularly scrutinize the inevitable impact of errors present in any realistic implementation. Importantly, we find that the hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, which opens the door to reach nontrivial time scales with a small number of gates. While errors within the random circuit are shown to be irrelevant, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware. Our work emphasizes the practical relevance of random circuits on NISQ devices beyond the abstract sampling task.
Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.
Abstract In the noisy intermediate-scale quantum (NISQ) era, quantum computers have become important tools to simulate the real-time evolution of strong correlated many-body systems. The general scheme of designing quantum circuits to simulate the quantum dynamics is based on the Trotter-Suzuki decomposition technique, which has its limitation because the accuracy of evolution process depends on the size of trotter steps when the evolution operator is decomposed as quantum gates. Going beyond the limitation of the Trotter-Suzuki scheme, we design a parameterized quantum circuit named α -circuit with simple determined size and the only one control parameter θ to simulate the real-time evolution of the specific XXX Heisenberg model with the specific initial state . The α -circuit can accurately generate the time-evolution results by tuning parameter θ , which means the circuit can also be regarded as a good state preparation machine (SPM).
The transport of conserved quantities like spin and charge is fundamental to characterizing the behavior of quantum many-body systems. Numerically simulating such dynamics is generically challenging, which motivates the consideration of quantum computing strategies. However, the relatively high gate errors and limited coherence times of today's quantum computers pose their own challenge, highlighting the need to be frugal with quantum resources. In this work we report simulations on quantum hardware of infinite-temperature energy transport in the mixed-field Ising chain, a paradigmatic many-body system that can exhibit a range of transport behaviors at intermediate times. We consider a chain with L=12 sites and find results broadly consistent with those from ideal circuit simulators over 90 Trotter steps, containing up to 990 entangling gates. To obtain these results, we use two key problem-tailored insights. First, we identify a convenient basis – the Pauli Y basis – in which to sample the infinite-temperature trace and provide theoretical and numerical justifications for its efficiency relative to, e.g., the computational basis. Second, in addition to a variety of problem-agnostic error mitigation strategies, we employ a renormalization strategy that compensates for global nonconservation of energy due to device noise. We discuss the applicability of the proposed sampling approach beyond the mixed-field Ising chain and formulate a variational method to search for a sampling basis with small sample-to-sample fluctuations for an arbitrary Hamiltonian. This opens the door to applying these techniques in more general models.
We investigate how to simulate a quantum many body system using near term device known as noisy intermediate scale quantum (NISQ) hardware. We explore the two different types of hardware, analog and digital device, and discuss their potential for quantum many body simulation. For instance, one of the applications is to use the analog device to simulate the slow dynamics under a strong staggered field. However, noise is a major challenge for quantum hardware. Hence, we discuss the quantum error mitigation methods which are suitable for NISQ hardware. We apply the most commonly-used quantum error mitigation techniques to our many body simulation. We simulate quantum body scar dynamics using zero noise extrapolation techniques, Pauli twirling, and dynamical decoupling. We also calculate the corresponding correlation function using direct measurement without resorting to ancilla qubits. Furthermore, we use the same error mitigation techniques to explore dynamics of energy correlators. As additional error mitigation technique, we divide the noisy correlation results by their sum, which is ideally constant due to energy conservation. By doing so, we significantly improve our result. Finally, we consider another NISQ friendly quantum algorithm for imaginary time evolution. With enough samples from minimally entangled typical thermal state algorithms, we can obtain the thermal expectation value of observable and explore their thermodynamics. In chapter 2, we provide the background knowledge for the following 4 chapters. We introduce the rydberg atom system as the analog system and their special parameter regime for interesting physics. Moreover, we introduce the digital quantum hardware and trotterization that enable one to simulate many body dynamics on the quantum hardware. We also introduce the quantum error mitigation techniques to reduce the the impact of noise from the results. Finally, we introduce the variation quantum dynamics method for the dynamics simulation. In chapter 3, we explore the Rydberg blockade regime with a strong staggered field. With the strong staggered field, any computational basis states are thermalized. Here, we investigate the slow dynamics using a Schrieffer-Wolff (SW) transformation, which is similar to perturbation theory in calculating the different-order effective Hamiltonian in terms of energy scale. As numerical result shown, the naive Schrieffer-Wolff transformation is not enough to explain the some dynamics' relaxation time. Hence, here, we also consider another conservative quantity to perform the SW transformation. The corresponding analytical result agrees with the numerical result. In chapter 4, we simulate quantum many body scarred dynamics in IBM quantum superconducting hardware. We also use custom pulse gates, quantum error mitigation techniques, and quantum error suppression methods to help us handle the noise. The error mitigated result outperforms the raw result. With the all techniques, we also explore non-trivial correlation function which also manifest the long lived oscillation in the rydberg blockade regime. In chapter 5, we use the same quantum error mitigation techniques, which we used in ch. 5, to study the energy transport properties of the mixed Ising model at infinite temperature. Here, instead of using some random state to evaluate the necessary two point energy density correlations, we use Y basis product state sampling to evaluate them. Surprisingly, few samples are enough to estimate the correlation. In addition to the quantum error mitigation techniques, we also use the fact that sum of all correlation functions is constant to renormalize the noisy results. With this trick, we are able to extract an effective dynamical exponent, which enables us to classify the transport properties. In chapter 6, we combine a variational quantum algorithm and the minimally entangled typical thermal states approach to investigate the thermodynamics of 1+1-dimensional Z2 lattice gauge theory. In particular, with a confining field, the 1+1-dimensional Z2 lattice gauge theory cannot be solved exactly. We estimate the corresponding thermal particle number density and energy density to distinguish the confined and deconfined regime. Chapter 7 provides a short review of the main results described in the thesis and a discussion of future research directions that they motivate. We mainly give insight into the recent experimental breakthrough in quantum error correction. We also clarify that quantum error mitigation methods do not become totally useless when a fault-tolerant quantum computer is realized. Finally, we also comment on variational quantum algorithms and their potential applications.
Noise is the defining feature of the NISQ era, but it remains unclear if noisy quantum devices are capable of quantum speedups. Quantum supremacy experiments have been a major step forward, but gaps remain between the theory behind these experiments and their actual implementations. In this work we initiate the study of the complexity of quantum random circuit sampling experiments with realistic amounts of noise. Actual quantum supremacy experiments have high levels of uncorrected noise and exponentially decaying fidelities. It is natural to ask if there is any signal of exponential complexity in these highly noisy devices. Surprisingly, we show that it remains hard to compute the output probabilities of noisy random quantum circuits without error correction. More formally, so long as the noise rate of the device is below the error detection threshold, we show it is #P-hard to compute the output probabilities of random circuits with a constant rate of noise per gate. This hardness persists even though these probabilities are exponentially close to uniform. Therefore the small deviations away from uniformity are hard to compute, formalizing an important intuition behind Google's supremacy claim. Interestingly these hardness results also have implications for the complexity of experiments in a low-noise setting. The issue here is that prior hardness results for computing output proba-bilities of random circuits are not robust enough to imprecision to connect with the Stockmeyer argument for hardness of sampling from circuits with constant fidelity. We exponentially improve the robustness of prior results to imprecision, both in the cases of Random Circuit Sampling and BosonSampling. In the latter case we bring the proven hardness within a constant factor in the exponent of the robustness required for hardness of sampling for the first time. We then show that our results are in tension with one another - the high-noise result implies the low-noise result is essentially optimal, even with generalizations of our techniques.