Circuits Learn Ordered States Beyond Equilibrium’s Limits

Scientists are increasingly exploring how to engineer novel phases of matter beyond the limitations imposed by conventional equilibrium physics. Fabian Ballar Trigueros and Markus Heyl, from Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, working with colleagues at the Centre for Advanced Analytics and Predictive Sciences (CAAPS) at the same institution, demonstrate that quantum circuits can learn states exhibiting long-range order inaccessible in equilibrium systems. Their research reveals the construction of variational circuits capable of generating symmetry-broken and symmetry-protected topological states in one dimension, even at finite energy densities where equilibrium states are typically unremarkable. This work is significant because it establishes coherent dynamics as a powerful tool for creating nonequilibrium phases, potentially broadening our understanding of order and its emergence in physical systems.

Can we create order in materials where conventional physics says it’s impossible. Yes, according to new work demonstrating that quantum circuits can learn states exhibiting long-range order. Even in one-dimensional systems where such order is normally forbidden. This opens up possibilities for designing materials with enhanced properties beyond those found in nature.

Scientists are increasingly interested in understanding phases of matter that exist far from equilibrium conditions. Equilibrium statistical ensembles, the traditional framework for describing these phases, impose strict limitations on what is possible, such as the Mermin-Wagner theorem which prohibits long-range order in low-dimensional systems beyond the ground state.

Research now demonstrates that quantum circuits can learn states of matter exhibiting long-range order inaccessible within the confines of equilibrium physics. Variational quantum circuits were constructed to generate symmetry-broken and symmetry-protected topological states with long-range order in one-dimensional systems, even at finite energy density where equilibrium states are typically unremarkable.

These learned states possess unconventional properties, displaying enhanced metrological characteristics akin to a GHZ state, while maintaining stability against local measurements. Beyond simply achieving order, The project establishes coherent quantum dynamics as a powerful tool for engineering novel, nonequilibrium phases of matter, potentially broadening the scope of quantum order beyond the usual constraints.

Unlike classical systems where non-equilibrium phases are well-documented, ranging from flocking behaviours to spontaneous pattern formation, the quantum area has lagged behind in establishing a general framework for such phases. Previous breakthroughs in areas like many-body localization and discrete time crystals suggest the possibility of extending order beyond equilibrium.

A unifying principle has remained elusive, particularly for ordered phases resembling those found in equilibrium statistical mechanics. Instead of being limited by these restrictions, the circuits explored here offer a pathway to engineer states that circumvent the limitations imposed by the Eigenstate Thermalization Hypothesis (ETH), which typically predicts featureless states at finite energy density in one dimension.

Numerical evidence indicates that coherent superpositions of individually featureless eigenstates, prepared by these variational circuits, can display macroscopic order. This approach lies in the construction of symmetry-constrained variational quantum circuits, employing a brickwork pattern of depth d with local k-qubit gates arranged across N qubits.

Each gate carries a variational parameter, providing the necessary degrees of freedom to steer the quantum state towards a desired configuration. The resulting ordered superposition states exhibit strong quantum signatures, with their quantum Fisher information scaling extensively and approaching the Heisenberg limit, indicating substantial multi-partite entanglement.

Long-range order emerges from spectral localisation despite random initial conditions

Once optimisation commenced, susceptibility values, denoted as χ, demonstrably saturated to a finite value — this value, while remaining below the theoretical maximum of 1, exhibited either approximate constancy or a slight increase alongside growing system size, N. Indicating the development of long-range order. Initial states employed were random product states, constructed with polar angles αi uniformly distributed between 0 and π. Ensuring unbiasedness with respect to the Z2 symmetry.

Performing independent optimisation runs with multiple random initializations and averaging observables ensured statistical robustness of Outcomes. Here, this observed order did not stem from artificial energy selection. Spectral analysis revealed a sharply localized support of the trained wave functions |ψθ∗⟩ within the energy eigenbasis of the reference Hamiltonian, centering around the spectrum’s midpoint.

At these energy densities, individual eigenstates were highly entangled but featureless, carrying negligible χ, aligning with expectations from the Eigenstate Thermalization Hypothesis (ETH). By according to ETH, matrix elements of local observables scale as O(E)δmn + e−S(E)/2f(Em, En). Where S(E) represents the thermodynamic entropy at mean energy E. The finite order observed originated from a coherent quantum superposition of numerous eigenstates.

In turn, the variational circuit amplified the collective contribution of ETH-suppressed off-diagonal matrix elements, generating macroscopic order. For a chosen reference Hamiltonian, H = −P i ZiZi+1 + 1 2ZiZi+2 −h P i Xi, the optimisation process minimised a variational objective L(θ) = −⟨χ⟩θ + σ⟨H −E⟩2 θ + β ⟨H2⟩θ −⟨H⟩2 θ, where σ and β are positive hyperparameters.

When considering the symmetry-protected topological (SPT) case, circuits discovered emergent symmetries leading to characteristic spectral degeneracies. Unlike the Landau case where optimised layers concentrated near Clifford gates, displaying near-integrable level statistics, the SPT circuits revealed a different behaviour. These circuits evaded the constraints of ETH, displaying clear signatures of nonergodicity. Stabilising long-range order at finite energy density in one dimension.

Variational circuit design and nonlocal string order parameter quantification

A constant-depth variational circuit, composed of three-qubit cluster-type operators and two-qubit XX rotations arranged in a brickwork pattern. Served as the central tool in this effort. Such circuits were designed to generate symmetry-broken and symmetry-protected topological (SPT) states in one-dimensional systems — by initialising the system within the ground state of a cluster state, a resource state frequently used in measurement-based quantum computation. Provided a starting point exhibiting Z2 × Z2 SPT order.

Then, the variational circuit was applied, with its parameters optimised to create highly excited states retaining nonlocal string order, a hallmark of SPT phases. Quantifying the emergence of topological order required a specific approach. Scientists introduced a nonlocal string order parameter, defined as Oij = Zi Yi+1 Qj−2 k=i+2 Xk Yj−1 Zj. To capture the characteristic correlations of the SPT phase across distant sites.

By evaluating this operator in the bulk of the system, specifically at i = N/4 and j = 3N/4 for a system of N qubits, they aimed to minimise boundary effects and isolate intrinsic topological correlations. Once the circuit parameters were optimised, The effort assessed convergence by monitoring the string order parameter’s value across training epochs for varying framework sizes.

To constrain the energy density and provide a reference point, the cluster, Ising model with open boundary conditions was employed as a Hamiltonian. At the same time, this Hamiltonian, defined as H = − N−1X i=2 Zi−1XiZi+1−Γ N−1X i=1XiXi+1−Γ 2 N X i=1Xi, allows interpolation between a pure cluster model. Realising Z2 × Z2 SPT order, and an Ising-like phase. Through adjusting the parameter Γ, The project team targeted highly excited states with nontrivial topological correlations.

For further analysis, the trained effective Hamiltonian was examined through spectral analysis, revealing four-fold degeneracies in each eigenstate. At the same time, histograms of trained weights were generated to understand the circuit’s behaviour and identify regions where the gates approached Clifford transformations. Such combined techniques allowed for a detailed investigation into how coherent dynamics can engineer nonequilibrium phases of matter beyond the limitations of equilibrium ensembles.

Quantum circuits overcome fundamental limits to create novel ordered states

Meanwhile, the creation of ordered states in systems defying conventional equilibrium physics is now within reach, as demonstrated by recent advances in utilising quantum circuits. Across decades, the Mermin-Wagner theorem has acted as a barrier, dictating that low-dimensional systems cannot sustain long-range order at energies above the ground state. This effort bypasses that restriction, showing that artificially driven, or ‘out-of-equilibrium’, frameworks can exhibit order previously thought unattainable.

This isn’t merely a refinement of existing techniques. It’s a departure from the established rules governing how order emerges in physical systems. The implications extend beyond fundamental physics, potentially reshaping our approach to quantum materials design. By employing variational circuits, in effect programmable quantum processors, researchers have engineered states with properties exceeding those found in static, equilibrium materials.

Instead of searching for materials that naturally exhibit desired order, this method proposes building order directly into a system’s dynamics. The scale of these circuits remains limited. Maintaining coherence, the fragile quantum state necessary for these computations, presents a considerable challenge. The ability to create states resistant to local measurements is particularly compelling.

Unlike many delicate quantum phenomena, these learned states appear to be surprisingly stable. Potential applications in quantum information processing. The broader effort will likely focus on scaling these circuits and exploring more complex forms of order — rather than simply replicating known phases of matter, the real promise lies in discovering entirely new, dynamically-driven states with properties we haven’t yet imagined. Where equilibrium physics defines the boundaries of what’s possible, these circuits suggest a far wider scope for engineering order in the quantum world.