Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer

The study targets challenging but feasible quantum simulation problems on current NISQ devices, focusing on strongly correlated condensed-matter systems where quantum criticality plays a central role. Quantum critical points feature diverging correlation lengths that cause strong finite-size effects, hindering both classical and quantum simulations. Tensor network methods, especially matrix product states (MPS) operating directly in the thermodynamic limit, offer a way to avoid finite-size scaling and can be translated into quantum circuits with potential quantum advantage. The purpose is to demonstrate that translationally invariant iMPS-inspired sequential circuits can accurately simulate both groundstate properties and non-equilibrium dynamics (including dynamical quantum phase transitions) of the transverse-field Ising model on Google's Rainbow (Sycamore-architecture) superconducting processor. The work aims to provide efficient circuit constructions and robust error mitigation strategies enabling accurate optimisation and time evolution at and across quantum criticality, highlighting the suitability of tensor-network-guided approaches for NISQ-era quantum advantage.

The paper builds on extensive prior work in tensor networks and quantum criticality. MPS and related tensor-network techniques provide state-of-the-art classical simulations for many spin systems and can be directly mapped to quantum circuits, potentially offering quantum advantage due to more favorable scaling with bond dimension (refs. 5–12, 33–35, 49). Quantum criticality and its role in condensed matter have been extensively reviewed (refs. 7–10). Prior proposals have used translationally invariant iMPS on quantum devices to evaluate observables via finite circuits using environment tensors determined by fixed-point equations (ref. 17). Recent advances include combining tensor networks with machine learning for phase identification and model compression (refs. 18–20, 50–52), and MPO-based error mitigation aligned with tensor-network structures (refs. 21, 22). Extensions to higher-dimensional sequential circuits and isometric PEPS-like constructions have been proposed (refs. 23, 24). For non-equilibrium dynamics, time-dependent variational principles and variational Trotterized updates for parameterized circuits provide efficient evolution strategies (refs. 38–43). The present work simplifies earlier time-evolution circuit proposals and integrates practical error mitigation tailored to NISQ hardware.

Model and ansatz: The transverse-field Ising Hamiltonian H = Σ_i [J Z_i Z_{i+1} + g X_i] is studied. The quantum state is represented by a translationally invariant iMPS-inspired sequential circuit with bond dimension D=2, implemented as a staircase of two-qubit unitaries U. Local observables of the infinite circuit are evaluated on finite-depth, finite-width circuits by introducing an environment unitary V = V(U) that captures the effect of the infinite environment. Environment determination and fixed-point equations: V(U) is obtained by solving fixed-point equations corresponding to the dominant eigenvectors of a transfer matrix associated with the iMPS. The cost includes a trace-distance-like penalty enforcing consistency between U and V. Circuits implement the three terms needed to compute Tr(P P_γ) (trace norms and cross terms), realized via swap tests, as detailed in Fig. 1 and Methods (Fig. 6). Groundstate optimisation: The total cost function C(θ) = ⟨ψ(θ)|H|ψ(θ)⟩ + Tr[ρ_i] − 2 Tr[ρ_i ρ_j^†] + Tr[ρ_j] is minimized using SPSA. Optimisation proceeds quasi-adiabatically in g: starting deep in a phase, parameters optimised at one g serve as initialisation for a nearby g, gradually approaching the quantum critical point. Circuits for the three Hamiltonian terms and the fixed-point consistency terms are executed in parallel on selected qubits. Overlap estimation via power method: Overlaps of translationally invariant states |U_A⟩ and |U_B⟩ are extracted from the principal eigenvalue λ of the transfer matrix 𝔈_{U_A U_B} using a power-method-inspired circuit evaluating C_n, with approximations to the top and bottom fixed points (β, τ). Choosing good β, τ accelerates convergence (exact fixed points yield n=1). Loschmidt echo normalisation C(U_A, U_A) mitigates depolarisation. Time evolution: Time updates are computed by maximising the overlap U(t+dt) = arg max_W |⟨W|e^{-iH dt}|U(t)⟩| within the ansatz. The time-evolution operator is Trotterised; a first-order scheme is used for practical circuit depth. Exploiting translational invariance, circuits act on even bonds only; projection back to a translationally invariant ansatz suppresses naive first-order Trotter errors (TDVP arguments imply equivalence to evolution with H/2 on even/odd partitions). The cost function uses a transfer-matrix power-method circuit (two powers shown) with approximations to fixed points accurate to O(Δt), yielding an estimate of the square of the principal eigenvalue up to O(Δt^2). Factorisations of both the state U and two-site Trotter unitaries to the Rainbow gate set minimise depth. Error mitigation and hardware strategies: (1) Qubit selection via empirical benchmarking using Loschmidt echo of representative circuits to identify highest-fidelity connected subgraphs; (2) Readout bias correction via a learned 2^M confusion matrix (M≤2 measured qubits here) applied to measurement outcomes; (3) Depolarisation mitigation via Loschmidt echo rescaling—circuits are overlapped with their Hermitian conjugates to estimate the depolarisation-induced rescaling; for dynamics, a near-identity time-evolution is included to match circuit structure; (4) Averaging results over four parallel circuit instances to reduce stochastic noise. Floquet calibration of native two-qubit gates was tested but provided negligible benefit compared to confusion-matrix correction and Loschmidt rescaling, and was not used in final results. Implementation details: Experiments were performed on Google’s Rainbow device (Sycamore architecture). D=2 ansatz with reduced two-qubit parameterisation was used; higher D remains possible via O(log D) depth circuits but was not required within present resolution. Optimisation employed SPSA with staged updates in g. Overlap circuits used n up to 6 in the power method; optimal balance of convergence and noise was found at n≈4–5.

- Groundstate accuracy: With depolarisation rescaling and readout correction, optimised energies on Rainbow closely match exact-in-ansatz D=2 values across g, including at the critical point g=1; except for g=0.4 (affected by device oscillations), rescaled energies are within 2.2% of exact-in-ansatz values. - Ansatz sufficiency: For the Ising model, D=2 performs remarkably well; going to D=4 yields improvements below current experimental resolution. Larger improvements are expected for other models (e.g., antiferromagnetic Heisenberg). - Overlap estimation: Loschmidt-corrected overlap estimates converge with power-method order n; n=4–5 achieves convergence within error bars. An outlier at n=6 was traced to an overestimate of the Loschmidt echo and corrected by interpolation. - Robust optimisation: SPSA successfully optimises U and V concurrently, with quasi-adiabatic stepping in g. A persistent half-hour oscillation in device performance notably impacted measurements around g=0.4. - Time-evolution fidelity: The simplified time-evolution cost function (even-bond Trotterisation, power-method transfer matrix) tracks exact dynamics in noiseless simulations and aligns, after rescaling, with on-device measurements. The optimal cost-function location along test parameter interpolations coincides with classical noiseless predictions at each time step. - Dynamical quantum phase transitions: Starting from the groundstate at g=1.5 and evolving with g=0.2, the logarithmic Loschmidt echo −log|⟨ψ(0)|ψ(t)⟩| exhibits periodic partial revivals and cusps characteristic of dynamical quantum phase transitions, captured by both simulations and on-device cost-function evaluations. - Circuit efficiency: The proposed circuits for overlaps and time evolution are considerably simpler than prior proposals while remaining effective under NISQ constraints.

The work addresses whether NISQ devices, guided by tensor-network structures, can simulate groundstate and dynamical properties of quantum critical systems directly in the thermodynamic limit. By employing translationally invariant iMPS-inspired circuits and environment tensors determined via fixed-point equations, the approach avoids finite-size scaling effects that hinder near-critical simulations. The results show that, with tailored error mitigation (confusion matrix for readout, Loschmidt echo rescaling for depolarisation, and careful qubit selection), D=2 iMPS circuits suffice to reproduce groundstate energies near the quantum critical point and to track real-time dynamics across a dynamical quantum phase transition. The cost-function based on the principal eigenvalue of a transfer matrix (estimated via a power method) provides a practical objective for both groundstate optimisation and time evolution, enabling shallow circuits tuned to device limitations. These findings highlight the compatibility of tensor-network variational principles with current superconducting hardware and suggest a pathway toward quantum advantage: increasing bond order and refining circuits could yield exponential gains relative to classical methods whose costs scale polynomially with bond dimension. The significance lies in demonstrating faithful dynamics and accurate groundstate energies at criticality on real hardware, validating tensor-network-informed, error-mitigated quantum simulation as a promising NISQ strategy.

This work demonstrates that translationally invariant, iMPS-inspired sequential circuits can simulate both groundstate properties and real-time dynamics of the transverse-field Ising model at and across quantum criticality on a superconducting quantum processor. Operating directly in the thermodynamic limit, the method avoids finite-size scaling issues and, with efficient error mitigation, achieves groundstate energies within 2.2% of exact-in-ansatz values and faithfully captures dynamical quantum phase transition signatures. The time-evolution cost function based on transfer-matrix principal eigenvalues is substantially simpler than prior proposals yet effective on NISQ hardware. Future directions include: (i) full on-device stochastic optimisation of the time-evolution cost function; (ii) increasing bond dimension using low-depth O(log D) circuits to enhance accuracy and explore regimes with potential quantum advantage; (iii) extending to two-dimensional sequential/isometric tensor-network circuits; (iv) integrating tensor-network-based error mitigation (MPO models) and leveraging gauge averaging; and (v) combining with machine learning to assist optimisation and mitigate both algorithmic and hardware-induced errors.

- Device noise and drift: A half-hour oscillation in device performance induced anomalous errors (notably at g=0.4); results depend on careful qubit selection and may vary day-to-day. - Depolarisation and readout errors: Reliance on Loschmidt echo rescaling and confusion-matrix readout correction is necessary; deeper circuits (higher power-method order or higher-order Trotterisations) amplify errors and uncertainty. - Ansatz and parameterisation constraints: A reduced two-qubit parameterisation for U leads to small systematic deviations from analytically exact results at large g; only D=2 was used experimentally, limiting expressivity. - Scalability of measurement correction: Confusion matrices scale as 2^M with measured qubits; although M≤2 here, larger M or higher bond dimensions may challenge scalability. - Trotter and projection errors: First-order Trotterisation and even-bond evolution introduce approximation errors; while mitigated by translational invariance/TDVP arguments, they remain finite. - Incomplete on-device optimisation: Time-evolution demonstrations used cost-function evaluation along interpolations rather than full on-chip stochastic optimisation at each step. - Circuit depth limits: Overlap estimation beyond n≈5 becomes noise-limited; higher bond dimension or more accurate fixed-point approximations may be constrained by depth/fidelity. - Floquet calibration was tested but not adopted due to limited benefit relative to other mitigation strategies.