Generation of Thermofield Double States and Critical Ground States with a Quantum Computer

THERMOFIELD DOUBLE STATES

• Tracing out system B from a TFD leaves system A in a thermal Gibbs state, enabling finite-temperature studies of the critical TFIM.The target is the TFIM at its critical point, g = 1, on a one-dimensional ring.Source: THERMOFIELD DOUBLE STATES · p. 2
• The protocol starts from Bell-pair singlets representing an infinite-temperature TFD, then variationally alternates inter-system and intra-system evolutions to target finite-temperature states.The parameters are selected to maximize fidelity with the target TFD.Source: THERMOFIELD DOUBLE STATES · p. 2
• The tailored four-operation ansatz avoids a full Trotterization of intra-system evolution because that could require too many steps, while extra B-system gates were unnecessary for high fidelity.The first two operations minimally Trotterize evolution with HA; the parameters are optimized classically for target-TFD fidelity.Source: THERMOFIELD DOUBLE STATES · p. 2
• Optimized TFD fidelities ranged from 0.93 at zero temperature to 1 at infinite temperature, and measured results agreed well with TFD expectations.Single-body observables and two-point correlation functions also compared well with the target TFD; imperfect entangling operations reduced correlations.Source: THERMOFIELD DOUBLE STATES · pp. 2–3
• At high temperatures, near-degenerate optimization can select large angles that increase errors, although the initial Bell pairs already approximate the target TFD well.In that regime, the passage notes that it would be better to avoid applying gates.Source: THERMOFIELD DOUBLE STATES · p. 3

QUANTUM CRITICAL STATE AT T = 0

• Long-range correlations and high entanglement challenge critical-state preparation, while finite-depth local circuits have finite correlation length and adiabatic digital circuits can be error-susceptible.Non-local gates may reduce the required steps, but tailoring trapped-ion power-law interactions to arbitrary critical states is difficult.Source: QUANTUM CRITICAL STATE AT T = 0 · p. 3
• The variational ansatz alternates HXX and HZ evolution, with parameters optimized to minimize energy for the ferromagnetic critical ground state.The ansatz is related to a Trotterized adiabatic sequence, for which sufficiently many layers can reach the target state.Source: QUANTUM CRITICAL STATE AT T = 0 · p. 4
• For optimal-angle protocols on the trapped ion system, measured two-body correlation functions for Pauli Z and X agree well with theory at p=1 but deviate at p=2 and p=3 as errors accumulate in deeper circuits.The energy calculated from correlation functions shows similar error accumulation; in experiment, the p=3 protocol generated states with energy as low as −5.46 ± −0.097.Source: QUANTUM CRITICAL STATE AT T = 0 · p. 4
• For seven qubits, the ideal p=1 and p=2 energies are −8.44 and −8.62, respectively, against a true ground-state energy of −8.98.The caption reports a gap of 0.23 and says it decreases linearly with system size.Source: Figure 3 · p. 5
• −8.02 ± 0.043 for p=1, versus −7.74 ± 0.095 for p=3; the lowest-depth protocol produced the lowest experimental energy.The simulated energies were −8.44 for p=1, −8.62 for p=2, and −8.98 for p=3, so the experimental ordering differed from the theoretical one.Source: QUANTUM CRITICAL STATE AT T = 0 · p. 5
• Using standard gradient descent in a hybrid QAOA loop that measures energy and adjusts parameters, optimization from random parameters for p=1 reached parameters differing from simulation, while measured energy matched theoretical prediction.For p=2, initialized at the numerical optimum, no cost-function drop was observed, so the results did not indicate a systematic-error shift away from the optimum.Source: QUANTUM CRITICAL STATE AT T = 0 · p. 5

Error simulation

• The model represents XX-gate angle error as θ = θ0(1 + Γr), where r is Gaussian noise, and averages results over 1000 samples.The modeled fractional angle error is Γ = 2δI/I, attributed to laser-intensity fluctuation.Source: Error simulation · p. 5
• The inferred Γ values are consistent between p=2 and p=3, and the predicted Γ and λ align with experimental accuracy.The paper notes that accounting for both experimental and theoretical errors matters as quantum computers and simulators scale up.Source: Error simulation · p. 6
• For Γ ≲ 0.13, higher-depth circuits perform better; at higher Γ, lower-depth circuits are preferable.The authors describe this as threshold behavior, with near-optimal solutions when noise remains below a critical value.Source: Error simulation · p. 6
• The depth threshold is present only for sufficiently low depolarizing noise λ; λ = 0.22 minimizes the variance of predicted Γ.The simulations vary λ from 0.1 to 0.3.Source: Error simulation · p. 6, Figure 4 · p. 6

• The critical TFIM ground-state preparation may support extracting universal quantum-critical properties, while TFD protocols could probe the finite-temperature quantum critical fan.Source: OUTLOOK · p. 6
• Verifying teleportation between the two sides of a TFD is proposed as an experimental test of wormhole traversal.Source: OUTLOOK · p. 6
• For thermal-state preparation, the hybrid approach would measure system-A free energy, requiring entanglement-entropy estimates from multiple Renyi measurements or randomized measurements.The authors suggest that hybrid preparation of pure and thermal states may eventually enable studies beyond classical-computer reach.Source: OUTLOOK · p. 6
• The experiment and noise analysis suggest that near-term devices must overcome an error threshold to unlock the full potential of variational approaches.Source: OUTLOOK · p. 6

• Residual entanglement between qubits and motional modes after a gate is modeled as depolarizing noise by averaging random-axis rotations after each XX gate.Source: Simulating depolarizing noise · p. 7
• A zero-mean Gaussian angle distribution with standard deviation λ is used, and averaging removes terms odd in the rotation angle.Source: Simulating depolarizing noise · p. 7
• Averaging the random rotations yields the single-qubit depolarizing channel, and the treatment extends to two qubits by rotating both after each XX gate.The channel parameter p is a function of λ.Source: Simulating depolarizing noise · p. 7

• The 171Yb+ ion-trap qubits use hyperfine ground states, optical pumping for initialization, fluorescence readout, and Raman-beam control.Reported typical fidelities are 99.5(2)% for single-qubit gates and 98−99% for two-qubit gates.Source: Experimental Details · p. 7
• Figure 5 plots measured one- and two-body correlators against theoretical expectations across target temperatures, grouping results by Pauli-X, -Y, and -Z.It includes intra-system and cross-system correlators; statistical error bars are no larger than the plotted symbols.Source: Figure 5 · p. 8
• Bell-pair singlets are initialized with RZ, RX, and XX gates, while QAOA unitaries use XX gates and single-qubit x-rotations to implement ZZ gates.Source: Experimental Details · p. 8

• For the periodic one-dimensional classical Ising model, the protocol uses a simplified sequence because of its extensive conserved quantities.The sequence is sufficient with p = L/2 iterations for L by 2 qubits.Source: Thermofield Double State of Classical… · p. 8
• For a six-qubit classical Ising system, one protocol iteration perfectly prepares the target TFD at any temperature.The optimal timestep parameters are obtained on a classical computer by maximizing the fidelity cost function.Source: Thermofield Double State of Classical… · p. 8
• TFD preparation is verified by measuring cross-system two-point correlators, intra-system correlators, and single-body observables.Source: Thermofield Double State of Classical… · p. 8

Symmetry Based Error Mitigation

• The TFIM Hamiltonian has a Z2 symmetry, which underlies the symmetry condition used for post-selection.Source: Symmetry Based Error Mitigation · p. 8
• The TFD symmetry requires Z1AZ2AZ3A = −Z1BZ2BZ3B, so measurements that violate this relation are discarded.Source: Symmetry Based Error Mitigation · p. 8
• Symmetry-based mitigation improves cross-system correlators, while the post-selection rate falls with temperature, consistent with larger errors at high temperature.The mitigation is applied to measurements in the Z direction.Source: Symmetry Based Error Mitigation · p. 8