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arXiv:2303.03268 · Physical Review Letters

Single-Qubit Error Mitigation by Simulating Non-Markovian Dynamics

Mirko Rossini, Dominik Maile, Joachim Ankerhold, Brecht Donvil

Published Sep 15, 2023• 12 citations
arXivHTMLPDF

Abstract

Quantum simulation is a powerful tool to study the properties of quantum systems. The dynamics of open quantum systems are often described by completely positive (CP) maps, for which several quantum simulation schemes exist. Such maps, however, represent only a subset of a larger class of maps: the general dynamical maps which are linear, Hermitian preserving, and trace preserving but not necessarily positivity preserving. Here we present a simulation scheme for these general dynamical maps, which occur when the underlying system-reservoir model undergoes entangling (and thus non-Markovian) dynamics. Such maps also arise as the inverse of CP maps, which are commonly used in error mitigation. We illustrate our simulation scheme on an IBM quantum processor, demonstrating its ability to recover the initial state of a Lindblad evolution. This paves the way for a novel form of quantum error mitigation. Our scheme only requires one ancilla qubit as an overhead and a small number of one and two qubit gates. Consequently, we expect it to be of practical use in near-term quantum devices.

DOI
10.1103/physrevlett.131.110603

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Quantum simulation is a powerful tool to study the properties of quantum systems. The dynamics of open quantum systems are often described by completely positive (CP) maps, for which several quantum simulation schemes exist. Such maps, however, represent only a subset of a larger class of maps: the general dynamical maps which are linear, Hermitian preserving, and trace preserving but not necessarily positivity preserving. Here we present a simulation scheme for these general dynamical maps, which occur when the underlying system-reservoir model undergoes entangling (and thus non-Markovian) dynamics. Such maps also arise as the inverse of CP maps, which are commonly used in error mitigation. We illustrate our simulation scheme on an IBM quantum processor, demonstrating its ability to recover the initial state of a Lindblad evolution. This paves the way for a novel form of quantum error mitigation. Our scheme only requires one ancilla qubit as an overhead and a small number of one and two qubit gates. Consequently, we expect it to be of practical use in near-term quantum devices.
While non-Hermitian Hamiltonians enable exotic dynamical phenomena, implementing their nonunitary time evolution on near-term quantum devices remains challenging. We propose a hardware-friendly protocol that simulates non-Hermitian dynamics without ancillas, controlled time evolution, or continuous monitoring. The protocol combines a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) evolution via classical Gaussian white-noise averaging with stochastic quantum error mitigation (QEM) to cancel the jump contribution at the level of expectation values. The mitigation layer uses only single-qubit operations. We validate the method through numerical simulations of an asymmetric-hopping model and an open {\it XXZ} spin chain with non-Hermitian boundary fields. Our work provides a programmable and ancilla-free framework for investigating exotic dynamics beyond the class of completely positive and trace-preserving dynamics using QEM.
Characterizing the memory properties of the environment has become critical for the high-fidelity control of qubits and other advanced quantum systems. However, current non-Markovian tomography techniques are either limited to discrete superoperators, or they employ machine learning methods, neither of which provide physical insight into the dynamics of the quantum system. To circumvent this limitation, we design learning architectures that explicitly encode physical constraints like the properties of completely-positive trace-preserving maps in a differential form. This method preserves the versatility of the machine learning approach without sacrificing the efficiency and fidelity of traditional parameter estimation methods. Our approach provides the physical interpretability that machine learning and opaque superoperators lack. Moreover, it is aware of the underlying continuous dynamics typically disregarded by superoperator-based tomography. This paradigm paves the way to noise-aware optimal quantum control and opens a path to exploiting the bath as a control and error mitigation resource.
Linear response theory and Green's functions provide a universal framework for understanding dynamical correlations in strongly correlated open quantum systems. While the theoretical foundation for non-Hermitian linear response has been recently established to describe dissipation and fluctuation-dissipation relations (FDR), generalizing these predictions onto practical quantum computers remains a formidable algorithmic challenge due to the intrinsically non-unitary nature of the dynamics. In this work, we present a systematic algorithmic framework that seamlessly transforms non-unitary multi-time correlation functions into a unitary form viable for digital quantum hardware. By mapping the vectorization of the Lindblad master equation into an expanded continuous-variable Liouville space, we employ the Schrödingerization technique to deterministically evaluate the non-Hermitian response. Furthermore, through hardware-aware simulations utilizing a 133-qubit device noise model, we demonstrate that our unitary framework robustly preserves the fundamental spectral information -- specifically the phase and oscillatory frequency -- against realistic depolarizing channels. By bypassing explicit non-unitary mid-circuit measurements, this approach intrinsically supports the integration of standard quantum error mitigation protocols, providing a scalable algorithmic blueprint for probing open-system universality on near-term hardware.
We develop randomized quantum algorithms to simulate quantum collision models, also known as repeated interaction schemes, which provide a rich framework to model various open-system dynamics. The underlying technique involves composing time evolutions of the total (system, bath, and interaction) Hamiltonian and intermittent tracing out the environment degrees of freedom. This results in a unified framework where any near-term Hamiltonian simulation algorithm can be incorporated to implement an arbitrary number of such collisions on early fault-tolerant quantum computers: we do not assume access to specialized oracles such as block encodings and minimize the number of ancilla qubits needed. In particular, using the correspondence between Lindbladian evolution and completely positive trace-preserving maps arising out of memoryless collisions, we provide an end-to-end quantum algorithm for simulating Lindbladian dynamics. For a system of nnn-qubits, we exhaustively compare the circuit depth needed to estimate the expectation value of an observable with respect to the reduced state of the system after time ttt while employing different near-term Hamiltonian simulation techniques, requiring at most n+2n+2n+2 qubits in all. We compare the cnot gate counts of the various approaches for estimating the transverse field magnetization of a 10-qubit XX-Heisenberg spin chain under amplitude damping. Finally, we also develop a framework to efficiently simulate an arbitrary number of memory-retaining collisions, i.e., where environments interact, leading to non-Markovian dynamics. Overall, our methods can leverage quantum collision models for both Markovian and non-Markovian dynamics on early fault-tolerant quantum computers, shedding light on the advantages and limitations of simulating open systems dynamics using this framework.
Quantum control has been a cornerstone of quantum information science, driving major advances in quantum computing, quantum communication, and quantum sensing. Over the years, it has enabled the implementation of arbitrary completely positive and trace-preserving (CPTP) maps; an important next step is to extend control to Hermitian-preserving and trace-preserving (HPTP) maps, which underpin applications such as entanglement detection, quantum error mitigation, quantum simulation, and quantum machine learning. Here we present an efficient and fully constructive method for implementing arbitrary HPTP maps. Unlike existing methods that decompose an HPTP map into multiple CPTP maps or approximate it using bipartite Hamiltonians with large Hilbert spaces, our approach compiles a target HPTP map into a single executable CPTP map whose Kraus rank is guaranteed to be no larger than the intrinsic rank of the target HPTP map plus one, followed by simple classical post-processing. Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings. Together with our numerical benchmarks, these results validate the efficiency and versatility of the proposed framework, opening a route to broader quantum-information applications enabled by HPTP processing.
Pauli channels are widely used to describe errors in quantum computers, particularly when noise is shaped via Pauli twirling. A common assumption is that such channels admit a Markovian generator, namely a Pauli-Lindblad model with non-negative rates, but the validity of this assumption has not been systematically examined. Here, using CP-indivisibility as our criterion for non-Markovianity, we study multi-qubit Pauli channels from a single snapshot of the dynamics. We find that while the generator always has the same structure as the standard Pauli-Lindblad model, the rates may be negative or complex. We show that random Pauli channels are almost always non-Markovian, with the probability of encountering a negative rate converging doubly exponentially to unity with the number of qubits. For physically motivated noise models shaped by Pauli twirling, including single-qubit over-rotations and two-qubit amplitude damping errors, we find that negative rates are generic, even when the underlying physical noise is Markovian. We generalize probabilistic error amplification and cancellation to non-Markovian generators, and quantify the sampling overhead introduced by negative and complex rates. Experiments on superconducting qubits confirm that allowing negative rates in the learned noise model yields more accurate predictions than restricting to non-negative rates.