Quantum error mitigation may simulate complex physics without extra qubits

Hiroki Kuji of Tokyo University of Science and colleagues have developed a new protocol that simulates complex physics governed by non-Hermitian Hamiltonians without requiring additional qubits. The work combines a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) evolution with stochastic quantum error mitigation (QEM) to model these systems on near-term quantum devices, a feat previously hindered by the need for ancilla qubits, controlled time evolution and continuous monitoring.

The error mitigation layer relies solely on single-qubit operations, offering a potentially scalable approach to investigating exotic dynamics beyond standard quantum mechanics. The researchers validated their method through simulations of asymmetric-hopping models and open XXZ spin chains.

Quantum Error Mitigation for General Non-Hermitian Dynamics

This approach allows for the simulation of non-Hermitian dynamics without relying on deep circuits requiring controlled unitary primitives or additional qubits, addressing a significant challenge for near-term quantum devices. A key feature of this work is the exclusive use of single-qubit operations within the mitigation layer, potentially enabling greater scalability compared to error correction schemes demanding multi-qubit operations.

The protocol constructs an auxiliary GKSL evolution and then cancels its quantum-jump contribution via QEM, a technique designed to eliminate the impact of errors at the level of expectation values. This cancellation is achieved through classical Gaussian white-noise averaging, further streamlining the implementation process and reducing hardware demands.

The researchers note that the sQEM quasi-probability normalization contributes a factor of Λ(T)^2 = exp(2κ T) to the required number of trajectories, a quantifiable metric for assessing the computational cost. The target non-Hermitian dynamics are defined by a Hamiltonian, $H_{target}$ = $H_{Re}$ + $i{H}_{Im}$, where $H_{Re}$ and $H_{Im}$ are Hermitian operators, and $H_{Im}$ is decomposed into local terms acting on finite sets of sites. This localized decomposition of the imaginary component of the Hamiltonian is important for the efficiency of the simulation.

The framework presented is programmable, offering a versatile approach to investigating exotic dynamics beyond those governed by completely positive and trace-preserving dynamics. “Quantum Error Mitigation Simulates General Non-Hermitian Dynamics,” states the paper’s headline, reflecting the core achievement of this hardware-friendly protocol.

The method’s success hinges on the ability to accurately simulate the GKSL evolution, which is achieved through discrete-time implementation and Gaussian noise averaging, as detailed in the supplemental material. This construction allows for the creation of a stochastic process that mimics the desired non-Hermitian dynamics, while the subsequent QEM step effectively suppresses the unwanted quantum jumps. By avoiding both ancilla overhead and the need for controlled time evolution, this protocol represents a step toward realizing complex quantum simulations on existing and near-future quantum hardware.

Stochastic Quantum Error Mitigation Cancels Jump Contributions

Simulations using 1.6 x ${10}^7$ trajectories demonstrate the protocol’s ability to accurately model quantum systems, as evidenced by comparisons to standard GKSL evolution without stochastic quantum error mitigation (sQEM). This level of fidelity is achieved by cancelling quantum jump contributions at the level of expectation values, an important step in simulating non-Hermitian dynamics. Researchers used B = 8000 independent runs, each averaging over $M_{num}$ = 2000 stochastic trajectories, to evaluate and starting from a specific initial state.

Data obtained without sQEM accurately reflect the corresponding GKSL evolution, including the jump contribution, providing a clear benchmark for the mitigation layer’s effectiveness. A minimal two-qubit validation, detailed in supplemental material, further confirms the accuracy of the method. Initializing the system in a product state |${\psi }_0$⟩ = |0110⟩, researchers monitored edge imbalance Δ $n_{edge}(t)$ = ⟨ to assess performance. This work provides a hardware-friendly route to non-Hermitian simulation.

Non-Hermitian Hamiltonian Decomposition into Real and Imaginary Parts

The decomposition of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator into non-Hermitian and jump components enables a new protocol for simulating quantum dynamics. This approach uses stochastic quantum error mitigation (sQEM) to specifically cancel the jump contribution within each computational trajectory, effectively realizing the target non-Hermitian Hamiltonian dynamics at the level of normalized expectation values. The final estimator is then derived through classical post-processing of measured outcomes, incorporating sQEM signs and normalization factors.

Engineered classical Gaussian white-noise processes, rather than uncontrolled device noise, form the basis of the noise-averaged evolution, where $E_f$[f_ℓ(t)] = 0 and $E_f$[f_ℓ(t)$f_{}$(ℓ^′)(t^′)] = 2γ_ℓδ_(ℓℓ^′)δ(t-t^′). The operators H_(I,ℓ) are constructed to reproduce the effective non-Hermitian contribution, $H_{Im}$, detailed in the End Matter section of the published work.

For nearest-neighbor interactions, an even-odd bond decomposition is applied to both the real and imaginary parts of the Hamiltonian. Averaging over the classical noise then yields an evolution that obeys the GKSL generator, L[ρ] = -i[$H_{Re},\rho$] + $su{m}_{}$(ℓ ∈ E)γ_ℓ(2H_(I,ℓ)ρH_(I,ℓ)-H_(I,ℓ)^2,ρ). Analysis of edge imbalance, Δ $n_{edge}(t)$ = ⟨, in a four-site chain is a benchmark for the full-bond non-Hermitian model. A shift, $E_{shift}I$, only rescales the overall norm and does not affect observable values relative to normalized time-evolved states.

Asymmetric-Hopping Model Validates the Mitigation Protocol

This methodology allows for the exploration of exotic physics without the need for ancilla overhead or controlled time evolution, a significant advantage for near-term quantum devices. The core of this advancement lies in the separation of the GKSL generator into a non-Hermitian Hamiltonian component and a jump contribution, with the latter systematically cancelled through stochastic QEM.

Researchers validated this framework by applying it to a one-dimensional interacting hard-core boson model featuring disordered asymmetric hopping, alongside an open XXZ spin chain with non-Hermitian boundary fields. The simulations demonstrate that the protocol accurately reproduces the target non-Hermitian dynamics even in scenarios with weak and strong disorder, as evidenced by comparisons to exact calculations.

This programmable, hardware-friendly route to simulating non-CPTP dynamics represents a departure from traditional methods, offering a pathway to investigate exotic phenomena previously inaccessible on current quantum hardware. “Our work establishes a programmable and hardware-friendly route to simulate exotic non-CPTP dynamics using QEM, without ancillas, controlled time evolution, or continuous monitoring,” the researchers state, highlighting the protocol’s potential to broaden the scope of quantum simulations. By using single-qubit operations within the mitigation layer, the method offers a potentially scalable solution for exploring complex physical systems and pushing the boundaries of quantum computation.

Open XXZ Spin Chain with Non-Hermitian Boundary Fields

Simulations of an open XXZ spin chain reveal a method for exploring non-Hermitian dynamics without the need for additional qubits, a feat previously hampered by the demands of maintaining quantum coherence. Researchers validated this approach by examining a model featuring non-Hermitian boundary fields, specifically an XXZ spin chain where interactions at the edges deviate from standard Hermitian behavior.

The work demonstrates that jump contributions, inherent in the simulation of non-Hermitian systems, can be systematically canceled using stochastic quantum error mitigation, a technique that uses classical computation to refine quantum results. This cancellation occurs at the level of expectation values, improving the accuracy of the simulation without increasing the complexity of the quantum circuit.

The number of jump maps canceled remains independent of the system size, L, resulting in a consistent cancellation rate even as the chain grows. “These results capture the non-Hermiticity-induced bias and its suppression by disorder,” the researchers report, providing qualitative validation of the protocol’s ability to model these complex systems.

The error bars associated with the Monte Carlo estimates remained comparable across different chain lengths, confirming the efficiency of the error mitigation strategy. This programmable route offers a departure from traditional methods requiring deep circuits and ancilla qubits, potentially accelerating progress in simulating exotic quantum phenomena.

Ancilla-Free Framework for Exotic Quantum Dynamics

The proposed protocol establishes a pathway to explore quantum dynamics beyond standard Hermitian and completely positive trace-preserving models, eliminating the need for additional ancilla qubits. The approach is particularly relevant given the limitations of near-term quantum devices, which struggle with the qubit overhead and complex operations typically required for accurate simulation. A key aspect of this work is the decomposition of the non-Hermitian Hamiltonian component into local terms, enabling a more manageable simulation process.

This localized approach facilitates the construction of stochastic generators used to simulate the target dynamics, offering a programmable route to investigate exotic physical phenomena. Instead, a classical processor samples Gaussian increments and stochastic QEM random variables, while the quantum device implements time evolution generated by the real part of the Hamiltonian and a stochastically driven unitary evolution.

This combination, coupled with the application of sampled QEM operations, yields final measurement outcomes that reflect the desired non-Hermitian behavior. The construction of the stochastic generators from the imaginary part of the Hamiltonian is an important step in realizing this broader applicability. The ability to simulate these systems without the constraints of traditional methods opens new avenues for exploring fundamental quantum phenomena and potentially designing novel quantum technologies. This approach offers a significant advantage in the pursuit of understanding and harnessing non-Hermitian physics.

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