A theory predicts power laws for errors during quantum annealing runs

A new field theory from Los Alamos National Laboratory predicts that the rate of errors during quantum annealing will follow specific power laws when occurring through or near a first-order phase transition.

Researchers Diego Andrés Rivera Orona, Predrag Punoševac, and Nikolai A. Sinitsyn developed this theory to model the emergence of defects during computations, anticipating sharp changes in the power exponents as continuous parameters are altered. Observation of these power law signatures would indicate first-order critical points, potentially enabling improved computational strategies. The driven Lipkin-Meshkov-Glick model demonstrates this behavior as a minimal model of interacting Ising spins. First-Order Phase Transition & Defect Production in Quantum Annealing This theory, developed at Los Alamos National Laboratory, forecasts that continuous changes in parameters will coincide with sharp shifts in the power exponents governing error rates, offering a potential method for identifying and circumventing problematic critical points. The work builds on the understanding that first-order transitions involve a metastable state, the decay of which generates abundant excitations, and provides a framework for predicting how these excitations manifest as errors. A constant offset, denoted as n0, appears in annealing experiments and captures an elevated production of excitations, potentially exhibiting only a slow logarithmic decay as the annealing process progresses. This observation aligns with similar defect scaling seen in ultracold chemical processes, suggesting a common underlying mechanism for excitation persistence even during slow annealing. Determining the size of this contribution requires extrapolating experimentally observed power laws to zero sweep rate, a technique that allows researchers to infer the magnitude of these persistent excitations and refine their understanding of the error landscape. The researchers note that stages of the first-order phase transition are key to understanding this behavior. To model this behavior, the team employed a real scalar φ4 theory with an external force, representing a coarse-grained spin polarization in strongly connected interaction networks. The Lagrangian of this field near the critical point is defined as L = mφ2/2–V(φ,t), where V(φ,t) = -β tφ-kφ^2/2 + gφ4/4, with m, g and β all greater than zero, and t representing time and β the sweep rate. This minimal model allows for the derivation of classical equations of motion describing the effective classical motion of the system, providing a bridge between dissipationless quantum phase transitions and classical field theories. The resulting equations, dx/dt = (∂ p) and dp/dt = -(∂ x), where Hcl(x,p) = p2/2m + V(x,t), describe the evolution of the system in phase space. Despite differing physical origins, the exponents 0. 75 and 0. 8, derived from the model, are remarkably close, allowing for a good fit to numerical data using a single power-law exponent supplemented by a logarithmic correction. This finding extends the Kibble-Zurek phenomenology, previously applied to thermal first-order phase transitions, to the realm of quantum annealing computations. The researchers demonstrated that the Suzuki-Zurek scaling can be inferred from quantum-coherent field theories, explaining its relevance to the observed behavior in quantum annealing. The findings align with observations from D-Wave experiments, which have reported unsuppressed contributions to the number of errors, suggesting that previous attempts to detect second-order phase transitions may have inadvertently captured first-order critical effects. Support for the presence of these first-order effects in D-Wave simulations also comes from observations of magnetic hysteresis, further strengthening the connection between the theoretical predictions and experimental results. “Therefore, it is likely that, by attempting to detect the scaling of a second-order phase transition, these experiments observed first-order critical effects,” the authors state. This suggests a refined approach to error mitigation in quantum annealing may involve actively identifying and avoiding these first-order critical points to improve computational performance.

Field Theory Modeling of Quantum Annealing Kinetics This work details how the rate of error generation scales with the speed at which a quantum annealing process traverses a first-order phase transition, predicting specific power laws governing these errors. Unlike models focused on second-order transitions, this theory accounts for the metastable states inherent in first-order systems, where the system can become temporarily trapped before settling into a ground state. The resulting error rate is a change in how those defects accumulate. Spatial dimensions become irrelevant in these networks, simplifying the field to a variable dependent only on time, φ(t). This simplification allows for the derivation of a minimal model, a one-dimensional quantum mechanical system where φ is treated as a coordinate of a particle moving in a time-dependent potential. The potential, V(x,t), is key to understanding the kinetics of defect formation, and its shape is dictated by parameters including the sweep rate, β, representing the speed of the transition. The researchers found that the field theory, described by the Lagrangian L = mφ2/2–V(φ,t), predicts a set of power laws governing the number of excitations, defects, created during the annealing process. This means the rate at which errors increase or decrease isn’t constant, but shifts predictably with changes in the annealing parameters.

The team demonstrated that the behavior is analogous to time-dependent second-order phase transitions, but with unique characteristics stemming from the metastable states. “Similar observations have been made in the theory of time-dependent second-order phase transitions,” the researchers write, noting that their model shares a common feature of integrability with other time-dependent field theories. The researchers’ approach uses the concept of “well-mixed” interactions, frequently found in quantum annealing computations due to the strongly connected graphs of spin-spin interactions used to define the problem. This allows them to move beyond specific system details and focus on the universal behavior near the phase transition. The work suggests a broader connection between algorithm performance and the universality classes of phase transitions, proposing that different scaling behaviors of defects could correspond to different convergence rates of computational algorithms. By establishing this correspondence, the team hopes to infer the existence of new, faster algorithms for certain computational problems. The ultimate goal, they suggest, is to characterize algorithm performance through the lens of quantum field theory, offering a new pathway to optimize computational efficiency.

Power Law Scaling of Error Rates with Sweep Rate A field theory predicts the number of defects created during quantum annealing scales according to specific power laws, dependent on the rate at which parameters are altered during computation. This prediction applies whether the process traverses or remains near a first-order phase transition, offering a new way to characterize error generation. The predicted scaling differs from that of second-order phase transitions, where the number of excitations, or defects, increases with a power law related to the sweep rate through the critical point. This new work accounts for a constant offset in defect production, even at very slow annealing speeds, suggesting a persistent source of errors during the process. The field theory derives a set of power laws describing the dependence of defect numbers not only on the sweep rate but also on the distance from the boundary between different quantum annealing regimes. The average number of excitations was calculated as the number of spins pointing opposite to their ground state direction as time approached infinity, confirming a power law for defect production in the subcritical regime. A density plot of excitation numbers as a function of sweep rate and spin-spin interaction parameter reveals the critical point is located at an interaction strength of 1, with excitation numbers remaining non-negligible even in the adiabatic limit, where the system should ideally remain in its ground state. The LMGm simulations demonstrate that the power law governing defect production terminates at a constant value as the sweep rate approaches zero, indicating a persistent level of excitation. This constant contribution exhibits an approximate power-law dependence on the distance between the interaction strength and its critical value. “The first-order transition can occur when unintentional energy barriers are introduced by small parameter uncertainties,” the theory explains, suggesting that these uncertainties can be suppressed to avoid unwanted critical points. The research indicates that minimizing these parameter uncertainties is important for achieving exponential suppression of excitations in the quasi-adiabatic regime, potentially leading to more reliable quantum computations. This is particularly relevant for computationally hard problems mapped onto Ising spin Hamiltonians, where finding the ground state is the ultimate goal. The ability to predict and potentially mitigate these error sources through careful control of annealing parameters represents a step toward improving the performance of quantum annealing algorithms. The findings suggest that well-mixed interactions within the system are key to deriving these universal behaviors and avoiding first-order critical points, ultimately enabling more efficient and accurate quantum computations.

Potential Barrier Dynamics During First-Order Transitions The dynamics of potential energy release during a first-order phase transition dictate the number of computational errors that emerge, according to a new field theory. Observation of this behavior would be a signature of first-order critical points and could help identify and avoid them for better computations. Reducing β suppresses. This scaling is not dangerous, since staying in the ground state requires a time that grows only as a power law, τ∝ Nα. How can excitations persist during very slow annealing?: Stages of the first-order phase transition. (a-c) Potential energy V(x) (solid curve) of the Hamiltonian, as in Eq.(24), at different times: (a) For t < 0, the system continuously follows a global minimum (black circle). (b) At t = 0, the initial minimum of V(x,t) is no longer global, but the potential barrier prevents the transition to a new global minimum. At the critical moment, t = tc, the initial local minimum of V(x,t) disappears, and the system then falls down the potential V(x,t), starting to oscillate around the global minimum. (d) A numerically computed trajectory of x(t). The emergence of high-amplitude oscillations corresponds to the transition to stage. Here, b = 0.002, and x(t) is obtained by solving Eq. (39). The research extends beyond simply identifying the presence of these errors. It also explores the conditions under which they become dominant. The study builds on previous work that considered dynamics with no potential barrier, allowing for the rescaling of variables to eliminate free parameters. This earlier research yielded a scaling exponent of 3/4, consistent with findings from the nonlinear Landau-Zener model in the critical regime. However, the current work emphasizes that the potential barrier becomes essential in the slowest regimes, influencing the dynamics near the critical time defined by a specific equation.

The team shifted variables to account for this critical time, allowing for a more accurate analysis of the system’s behavior. The work suggests that observing these power law signatures could enable the identification and avoidance of first-order critical points, ultimately improving computational performance.

The team’s findings highlight the importance of understanding the subtle interplay between transition speed and potential barrier dynamics in the context of quantum annealing, offering a pathway towards more robust and reliable quantum computations. The research provides a theoretical framework for predicting error rates and potentially mitigating these sources of error through careful control of annealing parameters.

Connecting Field Theory Parameters to Quantum Hamiltonian The connection between field theory parameters and the quantum Hamiltonian governing the Lipkin-Meshkov-Glick model (LMGm) reveals how the system’s mass, critical distance, and coupling strength relate to its behavior during quantum annealing. Specifically, the research establishes that the field Lagrangian, derived from the quantum model, is valid when the number of interacting spins is large and the Ising coupling approaches the critical value, where the coupling strength approximately equals the transverse magnetic field. This relationship is formalized in equations linking the parameters: m = 1/B_xS, k = (D-B_x)/S and g = Bx/2S³, where S represents the spin value. These equations provide a means to translate characteristics of the quantum Hamiltonian into parameters within the field theory, allowing for a more comprehensive analysis of the system’s dynamics. These laws are a mathematically defined relationship between these parameters and the rate at which errors accumulate. When the number of initial excitations, n₀, is small, the plus and minus velocities, ν + and ν, converge, indicating a consistent rate of excitation generation. However, in the ultra-adiabatic limit, a slight change in ν + occurs, suggesting a sensitivity to the annealing process itself. This detailed behavior is captured by the field theory’s ability to model the system as a one-dimensional quantum mechanical particle moving in a quartic potential. The potential energy, described by the equation Hcl(x,p,t) = p²/2m-kx²/2 + gx⁴/4-β tx, defines the landscape in which this particle moves, with parameters directly linked to the original quantum model. A canonical transformation, φ → -p and n → x, further simplifies the Hamiltonian, allowing for a clearer understanding of the forces acting on the particle. This transformation yields a classical particle moving in a quartic potential, enabling the researchers to analyze the system’s behavior using classical mechanics. The emergence of integrability in the effective field theories during the passage through a critical point also warrants further investigation, as the field dynamics near the stage, the point where the potential minimum becomes metastable, is described by the integrable Painlevé-I equation, a property not shared by the original quantum mechanical Hamiltonian. The study’s findings suggest that the LMGm provides a concrete system for testing the validity of the field theory. By analyzing the dynamics of interacting spins within the LMGm, researchers can observe the predicted power laws and gain insights into the underlying mechanisms driving error generation during quantum annealing. This understanding is important for identifying and potentially mitigating sources of error, ultimately improving the performance of quantum algorithms. 👉 More information🗞 Quantum annealing through a first-order phase transition: Field theory approach✍️ Diego Andrés Rivera Orona, Predrag Punoševac and Nikolai A. Sinitsyn🧠 DOI: http://link.aps.org/doi/10.1103/4945-qcb9 More like thisQuantum PhysicsQuantum systems may bypass slow driving with a new measurement shortcutPhysicsResidual coherence dominates entanglement in neutrino oscillation, study findsQuantum Error CorrectionA shifted operating point prevents controller stall in quantum error correctionPhysicsQuantum theorem shows black hole models are singularStay currentSee today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals. Tags: