Probing Deconfined Quantum Criticality Point on a Digital Quantum Computer

The gist The hardware resolves both ordered phases, and the crossing of the order parameters locates the transition in agreement with noiseless simulations and large-scale numerics<ref:2610.21761#pg6>.

Model and Phases

The study investigates a one-dimensional spin-1/2 chain described by a Hamiltonian with nearest-neighbour ferromagnetic and next-nearest neighbour antiferromagnetic interactions, which frustrates simple magnetic ordering and stabilizes distinct ordered phases<ref:2610.21761#pg4>. The competition between these terms drives a direct transition between a z-polarized ferromagnet (zFM) and a valence-bond solid (VBS)<ref:2610.21761#pg4>. The zFM phase breaks the spin-flip symmetry, characterized by uniform magnetization MFMz<ref:2610.21761#pg4>. Conversely, the VBS phase breaks translation symmetry and is characterized by the staggered bond modulation ΨVBS<ref:2610.21761#pg4>. The transition is driven by tuning the frustration parameter K across a range of 0.40 to 0.65<ref:2610.21761#pg4>.

Ground State Preparation and Circuit Implementation

The ground states are prepared using an ancilla-projected quantum imaginary-time evolution (QITE), which is compressed by tensor-network optimization into shallow depth quantum circuits that match the device connectivity<ref:2610.21761#pg4>. The process involves applying the enlarged unitary UQITE(β) to an initial system - ancilla state ψ0⟩ ⊗ 0a⟩, and then projecting onto the ancilla-0a⟩ branch using P0 = Isys ⊗ 0a⟩⟨0a to obtain the normalized projected QITE state<ref:2610.21761#pg4>. The circuit implementation uses a brickwork ansatz of depth d, consisting of single-qubit rotations and two-qubit entangling gates arranged according to the connectivity of the target IBM Quantum backend<ref:2610.21761#pg4>.

Hardware Results and Transition Location

The hardware successfully resolves both ordered phases, with the order parameters displaying a crossover between them as a function of K<ref:2610.21761#pg4>. The transition point Kc(L), defined by the crossing of the two order parameters MFMz(Kc) = ΨVBS(Kc), converges to a thermodynamic limit Kc(∞) = 0.555±0.003, which is consistent with noiseless simulations<ref:2610.21761#pg4>. The noise model based on device calibration accounts quantitatively for reduced amplitudes measured on hardware<ref:2610.21761#pg4>.

Characterization of Phases and Signatures

The two phases are characterized by spin and dimer correlations, structure factors, and correlation ratios<ref:2610.21761#pg4>. In the zFM phase (K = 0.40), the spin structure factor Sz(q) shows a pronounced peak at q = 0, indicating ferromagnetic correlations<ref:2610.21761#pg4>. At K = 0.65, the dimer structure factor SD(q) shows a sharp peak at q = π, signaling the staggered bond pattern of the VBS phase<ref:2610.21761#pg4>. The correlation ratio R becomes independent of L at a continuous transition, and for K in the critical region 0.53 ≲ K ≲ 0.57, RzFM falls while RVBS rises with increasing K<ref:2610.21761#pg4>.

Noise Analysis and Mitigation

The deviations from exact QITE are dominated by hardware noise, as shown by the noise model based on device calibration<ref:2610.21761#pg4>. The noise scale λ is determined by minimizing χ2(λ) between the IBMQ results and the simulated values<ref:2610.21761#pg4>. Compiling circuits against the exact projection probability keeps p0 large, which mitigates the effect of noise, leading to well-resolved VBS data for all chain lengths<ref:2610.21761#pg4>.

Conclusion

The approach combines exact target states, tensor-network circuit compression, and post-selected execution to study competing-order ground states on current quantum hardware<ref:2610.21761#pg4>. The framework offers a practical way to prepare and study competing-order ground states on current quantum hardware. The main limitation is the small projection probability of some circuits, which amplifies the effect of noise<ref:2610.21761#pg4>. Resolving the critical properties themselves will require larger systems, error mitigation beyond readout correction<ref:2610.21761#pg4>. The approach offers a practical way to prepare and study competing-order ground states on current quantum hardware.

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The correlation ratios of the two orders, which are largely insensitive to the noise-induced attenuation, reproduce on hardware the reversal of their size dependence across the transition<ref:2610.21761#pg4>. The measured snapshots show how the spin configurations change from ferromagnetic to dimerized across the transition<ref:2610.21761#pg4>. We also study noise model based on the device calibration shows that the deviations from exact QITE are dominated by hardware noise, and it reproduces the data in both phases when all error rates are scaled by a single factor λ ≈ 1.3–2 per run<ref:2610.21761#pg4>. The main limitation of the present framework is the small projection probability of some circuits, which amplifies the effect of noise; compiling the circuits against the exact projection probability keeps p0 large and gives well-resolved VBS data for all chain lengths<ref:2610.21761#pg4>. Resolving the critical properties themselves, such as the equal scaling of the two order parameters, will require larger systems, error mitigation beyond readout correction<ref:2610.21761#pg4>. Our approach, which combines exact target states, tensor-network circuit compression, and post-selected execution, thus offers a practical way to prepare and study competing-order ground states on current quantum hardware. A natural next step is to construct the target states from matrix-product-state calculations [39], so that the same circuit compression and post-selected execution on IBMQ can reach chain lengths beyond the reach of exact classical computation<ref:2610.21761#pg4>.

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The correlation functions and their ratios are averages over many measurements and do not reveal the individual configurations from which these averages arise<ref:2610.21761#pg4>. Such averages cannot distinguish, for example, an ordered state that is a superposition of oppositely ordered configura-tions from a disordered one, or a state in which each configuration is partially ordered from one composed of distinct ordered configurations<ref:2610.21761#pg4>. The Z-basis snapshots introduced in Sec. III resolve this, since each post-selected shot yields one spin configuration<ref:2610.21761#pg4>. Fig. 9 shows the distribution P(m, d) of the magnetization and dimerization of these shots for L = 12, at K = 0.40 (zFM), 0.55 (near the critical region), and 0.65 (VBS)<ref:2610.21761#pg4>. At K = 0.40, about half of the shots are fully polarized, at m = +1 in Fig. 9(a) and at both m = +1 and m = −1 in Fig. 9(d). In the latter case the average magnetization vanishes, although every shot is ferromagnetic; the snapshots thus reveal ferromagnetic order that the average conceals, and the boxes therefore give ⟨m⟩<ref:2610.21761#pg4>. At K = 0.55, fully polarized configurations coexist with weakly magnetized and partially dimerized ones, showing that the two orders compete shot by shot rather than through a uniform intermediate state<ref:2610.21761#pg4>. At K = 0.65, fully polarized shots have disappeared; the weight is centred at m ≈ 0 and shifted to d > 0, as shown in Fig. 9(f), reflecting the staggered dimer order of the VBS phase<ref:2610.21761#pg4>.

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The noise model also accounts for the correlation functions, as shown in Fig. 11<ref:2610.21761#pg4>. The spin correlations retain about 0.72 of their exact values on average and the dimer correlations about 0.84, consistent with the larger noise scale fitted to the spin run (λ = 1.82) than to the dimer run (λ = 1.38)<ref:2610.21761#pg4>.

1. Bold header: Post-Selected Quantum Imaginary-Time Evolution (QITE) for Ground State Preparation

This method allows digital quantum computers to prepare ground states of frustrated Hamiltonians with competing orders by using ancilla-projected QITE, which is compressed into shallow circuits that match the device connectivity. This enables the preparation of complex, non-trivial phases like the zFM and VBS states that are otherwise difficult to reach.

2. Bold header: Resolving Landau-Forbidden Transitions

The system can resolve this transition between two distinct symmetry-broken orders, such as a Landau-forbidden transition between competing orders, which is typically forbidden by the Landau-Ginzburg-Wilson paradigm. This capability is achieved by measuring both order parameters and showing the crossing of the order parameters locates the transition.

3. Bold header: Noise Model Calibration for Hardware Error Quantification

A noise model, based on the device calibration, can be used to account quantitatively for reduced amplitudes measured on hardware, allowing researchers to separate systematic circuit compression errors from device noise. This is achieved by fitting a scale factor λ to the simulated error rates of physical qubits.

4. Bold header: Resolving Phase Signatures via Correlation Ratios

The system can distinguish between phases in real space by analyzing correlation ratios, which are largely insensitive to the overall attenuation of the structure-factor peaks and are largely insensitive to a uniform reduction of the correlations. This allows for the resolution of the reversal of both ratios, [across the transition], indicating where one order is long-ranged and the other is short-ranged.

5. Bold header: Snapshot Distribution Analysis for Configuration Discrimination

The system can analyze snapshot distributions to distinguish between different configurations, showing how "the weight is centred at m ≈ 0 and shifted to d > 0, reflecting the staggered dimer order of the VBS phase." This capability reveals that states near criticality are not uniform but represent a superposition of distinct ordered configurations.