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Register a new userLow Rank Density Matrix Evolution for Noisy Quantum Circuits
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In this work, we present an efficient rank-compression approach for the classical simulation of Kraus decoherence channels in noisy quantum circuits. The approximation is achieved through iterative compression of the density matrix based on its leading eigenbasis during each simulation step without the need to store, manipulate, or diagonalize the full matrix. We implement this algorithm in an in-house simulator, and show that the low rank algorithm speeds up simulations by more than two orders of magnitude over an existing implementation of full rank simulator, and with negligible error in the target noise and final observables. Finally, we demonstrate the utility of the low rank method as applied to representative problems of interest by using the algorithm to speed-up noisy simulations of Grovers search algorithm and quantum chemistry solvers.
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Simulating quantum circuits with classical computers requires resources growing exponentially in terms of system size. Real quantum computer with noise, however, may be simulated polynomially with various methods considering different noise models. In this work, we simulate random quantum circuits in 1D with Matrix Product Density Operators (MPDO), for different noise models such as dephasing, depolarizing, and amplitude damping. We show that the method based on Matrix Product States (MPS) fails to approximate the noisy output quantum states for any of the noise models considered, while the MPDO method approximates them well. Compared with the method of Matrix Product Operators (MPO), the MPDO method reflects a clear physical picture of noise (with inner indices taking care of the noise simulation) and quantum entanglement (with bond indices taking care of two-qubit gate simulation). Consequently, in case of weak system noise, the resource cost of MPDO will be significantly less than that of the MPO due to a relatively small inner dimension needed for the simulation. In case of strong system noise, a relatively small bond dimension may be sufficient to simulate the noisy circuits, indicating a regime that the noise is large enough for an `easy classical simulation. Moreover, we propose a more effective tensor updates scheme with optimal truncations for both the inner and the bond dimensions, performed after each layer of the circuit, which enjoys a canonical form of the MPDO for improving simulation accuracy. With truncated inner dimension to a maximum value $kappa$ and bond dimension to a maximum value $chi$, the cost of our simulation scales as $sim NDkappa^3chi^3$, for an $N$-qubit circuit with depth $D$.
We study the fundamental design automation problem of equivalence checking in the NISQ (Noisy Intermediate-Scale Quantum) computing realm where quantum noise is present inevitably. The notion of approximate equivalence of (possibly noisy) quantum circuits is defined based on the Jamiolkowski fidelity which measures the average distance between output states of two super-operators when the input is chosen at random. By employing tensor network contraction, we present two algorithms, aiming at different situations where the number of noises varies, for computing the fidelity between an ideal quantum circuit and its noisy implementation. The effectiveness of our algorithms is demonstrated by experimenting on benchmarks of real NISQ circuits. When compared with the state-of-the-art implementation incorporated in Qiskit, experimental results show that the proposed algorithms outperform in both efficiency and scalability.
The study of the impact of noise on quantum circuits is especially relevant to guide the progress of Noisy Intermediate-Scale Quantum (NISQ) computing. In this paper, we address the pulse-level simulation of noisy quantum circuits with the Quantum Toolbox in Python (QuTiP). We introduce new tools in qutip-qip, QuTiPs quantum information processing package. These tools simulate quantum circuits at the pulse level, fully leveraging QuTiPs quantum dynamics solvers and control optimization features. We show how quantum circuits can be compiled on simulated processors, with control pulses acting on a target Hamiltonian that describes the unitary evolution of the physical qubits. Various types of noise can be introduced based on the physical model, e.g., by simulating the Lindblad density-matrix dynamics or Monte Carlo quantum trajectories. In particular, we allow for the definition of environment-induced decoherence at the processor level and include noise simulation at the level of control pulses. As an example, we consider the compilation of the Deutsch-Jozsa algorithm on a superconducting-qubit-based and a spin-chain-based processor, also using control optimization algorithms. We also reproduce experimental results on cross-talk noise in an ion-based processor, and show how a Ramsey experiment can be modeled with Lindblad dynamics. Finally, we show how to integrate these features with other software frameworks.
A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided through a central-limit theorem, which validity extends beyond the present context. It holds for example for estimations of the Jones polynomial. Interestingly, the kind of state preparations and measurements involved in this application can in principle be made instantaneous, i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accurately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allows one to reconstruct scattering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetizations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently.
Noise in existing quantum processors only enables an approximation to ideal quantum computation. However, these approximations can be vastly improved by error mitigation, for the computation of expectation values, as shown by small-scale experimental demonstrations. However, the practical scaling of these methods to larger system sizes remains unknown. Here, we demonstrate the utility of zero-noise extrapolation for relevant quantum circuits using up to 26 qubits, circuit depths of 60, and 1080 CNOT gates. We study the scaling of the method for canonical examples of product states and entangling Clifford circuits of increasing size, and extend it to the quench dynamics of 2-D Ising spin lattices with varying couplings. We show that the efficacy of the error mitigation is greatly enhanced by additional error suppression techniques and native gate decomposition that reduce the circuit time. By combining these methods, we demonstrate an accuracy in the approximate quantum simulation of the quench dynamics that surpasses the classical approximations obtained from a state-of-the-art 2-D tensor network method. These results reveal a path to a relevant quantum advantage with noisy, digital, quantum processors.
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