Crossing a topological phase transition with a quantum computer

• The Hamiltonian has trivial, symmetry-broken, and SPT phases under global spin-flip and time-reversal symmetries.The SPT phase contains the cluster state; the trivial phase contains the paramagnetic product state, and the symmetry-broken phase contains the Ising ferromagnetic ground state.Source: §I · p. 2, Figure 1 · p. 2
• The chosen path connects the cluster Hamiltonian at g = −1 to the trivial paramagnet at g = 1, with the transition at g = 0.The path is the black curve in the phase diagram.Source: §I · p. 2, Figure 1 · p. 2
• Non-local string order parameters distinguish the phases: in the infinite-length limit they are nonzero in one phase and zero in the other.Along this path, SZY(g) and S1(g) are zero on opposite sides of g = 0, and their nonzero values follow 4|g|/(1+|g|)^2.Source: §I · p. 2
• Along the selected path, the string-order values are independent of string length and equal their thermodynamic-limit values; this property holds only on that path.Away from the black line, macroscopic string lengths are generally needed to sharply distinguish the phases.Source: §I · p. 2

• The iterative ground-state construction for an infinite chain has an equivalent finite circuit for measuring observables with finite connected support.Measurements exclude the unphysical first and last qubits and match the corresponding infinite-chain measurements.Source: A. Infinite state as finite quantum… · p. 2, Figure 2 · p. 3
• The repeated circuit element away from the measured observable acts as a transfer matrix, whose dominant right eigenvector determines the thermodynamic-limit expectation value.U1 is chosen so its circuit vector is the transfer matrix's dominant right eigenvector with eigenvalue 1.Source: A. Infinite state as finite quantum… · p. 3, Figure 3 · p. 3
• The circuit elements below the measured operator cancel by unitarity, while the fixed-point circuit represents the remaining repeated elements above it.The dominant left eigenvector corresponds to the identity.Source: A. Infinite state as finite quantum… · p. 3

• Direct measurement reconstructs Pauli-string expectations from simultaneous measurements of relevant qubits, while interferometry extracts the string order parameter by measuring an entangled ancilla.Direct measurement uses shallower circuits, whereas interferometry requires only one-qubit readout.Source: §II · p. 3
• The simulations used IBM's 27-qubit Toronto device on October 25, 2021, with subsets of system qubits selected to favor CNOT fidelity while limiting readout error and coherence time.Direct-measurement data also received readout-error mitigation; each circuit was run 8192 times.Source: §II · pp. 3–4
• Average energy density is compared with the analytic result for N = 5–9 qubits, using central sites and excluding the end qubits.The figure compares device data with the analytic energy.Source: Figure 4 · p. 4

• The measured energy follows the exact functional form despite an absolute-value discrepancy, attributed mainly to dephasing and measurement error.Dephasing changes the curve's shape near g = 0 and is expected to increase with circuit depth and N; measurement error accounts for slight asymmetry across g = 0.Source: A. Direct Measurement · p. 4
• Direct string-order measurements qualitatively agree with analytics for shorter strings, but the transition is no longer detectable at N = 9 and l = 7.The order parameters identify the phase transition at shorter lengths, but that information is lost beyond approximately l = 6.Source: Figure 5 · p. 4, A. Direct Measurement · p. 4
• As string length increases, measurement accuracy declines because both the number of qubits and the measured non-local operator's length increase.The decline is more pronounced than for the energy-density measurements.Source: A. Direct Measurement · p. 4

B. Interferometry Experiment

• The ancilla’s measured expectation value gives the real part of the Pauli-string expectation, equivalent to the string order parameter.The ancilla controls the Pauli-string operation and is measured after a final Hadamard gate.Source: B. Interferometry Experiment · p. 5
• Interferometry improves quantitative accuracy over direct measurement and clearly reveals the phase transition for both string order parameters at all tested lengths.Measured values do not decrease monotonically with string length, suggesting run-to-run fluctuations are comparable to or larger than the added error from deeper circuits.Source: B. Interferometry Experiment · p. 5

III. ERROR MODELLING

• The direct method measures many qubits with shallow circuits, whereas interferometry measures one qubit using deeper circuits.These differences expose the methods to device errors differently.Source: §III · p. 5
• The analysis models measurement error, unitary gate error, and decoherence, plotting relative error against string length and the relevant error rate.The relative error is evaluated at g = 0.5 for S1 and g = −0.5 for SZY.Source: §III · p. 5

A. Measurement Error

• The model applies an independent bit flip with probability ϵ to each measured qubit, updates the bit-string probabilities, and evaluates string order from the updated distribution.The flip matrix depends on the number of bit flips between basis states.Source: A. Measurement Error · pp. 5–6
• Direct-measurement relative error increases with string length, while interferometry’s measurement error does not because it measures only one qubit.For the IBM device, the average readout error corresponds to approximately ϵ = 0.03.Source: A. Measurement Error · p. 6
• Approximately 30% error in the string order parameter at l = 5 can arise from readout error alone for the direct method, with larger errors for l > 5.The figure also compares error across string lengths and error rates for the modeled error sources.Source: A. Measurement Error · p. 6, Figure 8 · p. 6

B. Unitary Gate Error

• The model assumes perfect single-qubit gates and random unitary perturbations on entangling CNOT gates.The simulations average over 100 realizations of the random gates.Source: B. Unitary Gate Error · p. 6
• For both methods, error increases with string length; interferometry is more sensitive to unitary gate error because its circuit uses more gates.The difference between the methods is much less drastic than for measurement errors.Source: B. Unitary Gate Error · p. 6
• The IBM device’s unitary gate error cannot be quantified because it is difficult to separate from other decoherent error sources.Source: B. Unitary Gate Error · p. 7

C. Decoherence

• The depolarizing channel is applied just before measurement, although real decoherence occurs throughout circuit implementation, state preparation, and readout.The authors describe this timing as an approximation.Source: C. Decoherence · p. 7
• ϵ = (1 − e^−t/T) sets the mixing with a completely mixed state, with t estimated from asynchronous CNOT gates at 425 ns each and T approximately 100 µs.Single-qubit gates are assumed instantaneous.Source: C. Decoherence · p. 7
• The modeled depolarizing error grows approximately linearly with string length l, while the direct and interferometric methods differ by only two asynchronous CNOT layers.Other additional CNOT gates can be run in parallel with gates common to both methods.Source: C. Decoherence · p. 7
• The S1 circuit at g = 0.5 has one more CNOT than SZX at g = −0.5, and relative error varies exponentially with decoherence time.The comparison and exponential behavior are shown in Fig. 8(f).Source: C. Decoherence · p. 7

IV. DISCUSSION

• The studied path permits an especially efficient ground-state construction, while the authors propose that MPS can be constructed similarly and variationally optimized on quantum computers.The discussion also notes that finding optimal ansatz circuits for variational optimization remains open, and relates sequential circuits to sparse representations and possible higher-dimensional tensor-network advantages.Source: §IV · p. 7
• Interferometric string-order measurement agreed qualitatively with exact results, while readout errors dominated on IBM devices and favored its single-qubit measurement over direct multi-qubit measurement.Other technologies could have a different balance of measurement and gate errors, making direct measurement preferable; the authors stress tailoring methods to device errors.Source: §IV · p. 7

Appendix A: Connection to infinite MPS

• The infinite-MPS connection represents translationally invariant states with local entanglement using repeated finite-bond-dimension tensors and transfer-matrix eigenvectors, enabling finite-cost expectation values.The states considered use a single-site unit cell and bond dimension 2; right-canonical form makes the corresponding two-qubit gate unitary, and the connection is not restricted to bond dimension 2.Source: Appendix A · p. 8

1. Quantum circuit in elementary gates

• The appendix embeds canonical-form matrices into two-qubit unitaries and decomposes the circuit’s controlled-unitary gates into the IBM devices’ elementary gates.The sequences and angles were selected to match the matrix representations of U and U1.Source: §1 · p. 9, Figure 10 · p. 9

Appendix B: Error mitigation

• Readout mitigation applies the inverse or a suitable pseudo-inverse of a tomography-derived matrix, but is not scalable beyond accurately simulable device sizes.The matrix maps ideal basis states to measured outcome distributions; constructing it requires partial tomography.Source: Appendix B · pp. 9–10

1. Derivation

• The Hadamard test measures a Pauli-string expectation by applying a controlled unitary between ancilla Hadamard gates and reading the ancilla’s Z expectation.Because the Pauli string is Hermitian, its expectation is real.Source: §1 · p. 10

2. Circuit Decomposition

• Controlled Pauli strings are implemented with CNOTs and, for the second string form, H and S basis changes; swaps route the ancilla through nearest-neighbor qubits.Cancelling pairs of CNOTs simplifies the circuit, while parallel gates reduce asynchronous CNOT layers.Source: §2 · pp. 10–11, Figure 11 · p. 10

Appendix D: Dephasing

• The dephasing model assumes pure phase-flip errors, which alter X and Y but not Z components; the authors do not model dephasing separately for string-order parameters.They expect its effect there to resemble bit-flip error, with different length dependence for SZY and S1, and say their Section III conclusions would be unaffected.Source: Appendix D · p. 11
• The dephasing model reproduces the experimental energy-density curve’s dip at g = 0, and comparison with experiment indicates significant dephasing.The model’s plotted parameter p is related to dephasing rate and circuit depth.Source: Appendix D · p. 11, Figure 12 · p. 11