[challenge]: Criticality in open quantum matter

Released by

Guo-Yi Zhu, Hong Kong University of Science and Technology (Guangzhou)

Contact email

guoyizhu@hkust-gz.edu.cn

Method

Quantum Monte Carlo

Challenge issue

Released by

Guo-Yi Zhu, Hong Kong University of Science and Technology (Guangzhou)

Contact email

guoyizhu@hkust-gz.edu.cn

Method

Tensor network contraction / Monte Carlo sampling / finite-size scaling

Challenge issue

The rapid development of quantum processors has intensified the study of open quantum systems, where measurement and noise act alongside Hamiltonian dynamics. Monitored and decohered quantum states can exhibit unconventional phases and critical phenomena with no direct counterpart in Hamiltonian ground states. This challenge asks participants to reproduce universal conformal data for representative examples and then investigate more exotic, intrinsically quantum critical points.

Criticality in open quantum matter

At a continuous phase transition, the correlation length diverges and correlations become scale invariant. In many local systems, scale invariance enlarges to conformal symmetry, and the critical point is described by a conformal field theory (CFT). In $1+1$ spacetime dimensions, a CFT organizes universality through a compact set of conformal data, including scaling dimensions and the central charge $c$.

In unitary CFTs, the central charge measures the number of emergent gapless degrees of freedom: a Majorana mode has $c=\tfrac12$, while a free boson has $c=1$. It also controls finite-size and entanglement scaling and decreases along renormalization-group flows according to Zamolodchikov's $c$-theorem. Measuring it therefore provides a sharp fingerprint of a universality class. This challenge seeks the corresponding effective central charge in critical matter created by measurement and decoherence.

Where does such critical matter actually show up? Two broad settings have made it concrete:

• Monitored quantum circuits. Interleaving random unitary gates with local measurements can drive a measurement-induced phase transition (MIPT) between volume-law and area-law entangled phases. At the transition, the trajectory-resolved entanglement entropy grows logarithmically with subsystem size. Its conformal description was developed in arXiv:1908.08051, while transfer-matrix studies extracted an effective central charge and a tower of operator scaling dimensions in arXiv:2107.03393. More structured circuits can instead realize Nishimori transitions associated with measurement-based preparation of long-range-entangled states (arXiv:2208.11136, arXiv:2309.02863, and arXiv:2208.11699).
• Noisy or decohered quantum states. Exposing a topological memory such as the toric code to noise, or decohering a symmetry-protected or topological state, produces a mixed state whose criticality determines an error-correction or learning threshold (arXiv:quant-ph/0110143). Born-rule statistics map these problems to disordered classical models and $(1+1)$D non-unitary CFTs, including Nishimori and self-dual critical points in deformed toric codes and decohered cluster states (arXiv:2403.04767, arXiv:2504.12385, and arXiv:2502.14034).

What are the (effective) central charges for these open critical quantum states?

A monitored or decohered quantum system produces an ensemble of trajectories labelled by measurement records $m$. If $|\psi(m)\rangle$ denotes the corresponding unnormalized trajectory state, its Born weight is $\langle\psi(m)|\psi(m)\rangle$. Representing the circuit or state as a tensor network identifies this weight with a classical partition function $Z_m$ in the same spacetime dimensions. The normalized Born probability is therefore

$$
P(m)=\frac{Z_m}{\mathcal Z},\qquad \mathcal Z=\sum_m Z_m.
$$

The central quantity is the disorder-averaged free energy

$$
\boxed{F=-\sum_m P(m)\ln Z_m},
$$

which is contributed by the Shannon entropy of the measurement records and a background term. It is obtained by contracting the 2D tensor network for $Z_m$ and averaging over Born-sampled records. Writing $Z_m=tr(T_{L_{\tau}}\cdots T_1(m))$ as a product of random row-to-row transfer matrices, the free energy per row is governed by the largest Lyapunov exponent $\gamma_1(L)$. Its finite-size (Casimir) correction on a cylinder of circumference $L$ defines the effective central charge:

$$
\overline{\gamma_1(L)}= f_\infty L - \frac{\pi c_{\rm eff} \alpha}{6L}+\dots
$$

A robust calculation iterates the random transfer matrices with periodic QR reorthogonalization, averages $\gamma_1(L)$ over Born-sampled disorder, and fits the $1/L$ correction across several circumferences $L$.

The prefactor $\alpha$ is the effective sound velocity, or spacetime-anisotropy factor: it relates correlation lengths along the spatial and temporal directions of the network. Because $\alpha$ is generally non-universal, it must be calibrated independently by comparing spatial and temporal correlation functions at criticality, as in arXiv:2107.03393. A useful shortcut is to design the $(1+1)$D circuit so that the corresponding 2D classical model is spacetime-isotropic, giving $\alpha=1$ by construction.

The full Lyapunov spectrum contains more information: its low-lying levels form the finite-size CFT spectrum on a circle of circumference $L$,

$$
E_m \simeq \frac{2\pi\alpha}{L}\Big(\Delta_m-\frac{c_{\rm eff}}{12}\Big)+\dots,
$$

so the vacuum level determines $c_{\rm eff}$, while the gaps determine the operator scaling dimensions $\Delta_m$. See Fig. 13 of arXiv:2502.14034 for an example.

Goal

Reproduce the effective central charge $c_{\rm eff}\equiv c_{\rm Casimir}$ at three critical points and validate each result against an independently known value. Reproducing the central charge of an unstructured MIPT is optional.

Critical point	Target $c_{\rm eff}\equiv c_{\rm Casimir}$	Reference
Clean Ising (warm-up)	$1/2$	Textbook
Nishimori (noisy toric code / RBIM)	$0.464(4)$	arXiv:cond-mat/0010143
Weak self-dual (decohered toric code)	$0.447(1)$	arXiv:2502.14034

1. Warm-up: clean Ising. Reproduce $c=\tfrac12$ for the clean 2D Ising model, or equivalently the critical transverse-field Ising chain, from the leading transfer-matrix eigenvalue. This validates the Casimir fit.
2. Nishimori point. Map a noisy toric code to the $\pm J$ random-bond Ising model on the Nishimori line. Extract $c_{\rm eff}\equiv c_{\rm Casimir}$ from the disorder-averaged free energy and reproduce $0.464(4)$. A high-precision estimate of the critical point is given in arXiv:2511.02907.
3. Weak self-dual point. Map the decohered SPT or toric-code state with self-duality to its self-dual random-bond Ising model; a concise statistical-mechanics description is available in this KITP chalk talk. Extract $c_{\rm eff}\equiv c_{\rm Casimir}$ and reproduce $0.447(1)$. Use a spacetime-isotropic lattice so that $\alpha=1$. A free-Majorana representation reduces the Lyapunov-exponent calculation to a Chalker–Coddington-type network problem. But the disorder is highly correlated here due to the Born's rule, so one has to sample the tensor network first.
4. Open research challenge: learning-induced metal–insulator transition. In Learning transitions of topological surface codes (arXiv:2512.19786), projective measurement of a surface code in a generic uniform basis maps to a disordered free-fermion network model in symmetry class DIII. Its Majorana-metal phase undergoes a metal–insulator (learning) transition as the measurement basis is biased toward $X$ or $Z$. The central charge of this transition remains unknown. Locating the transition may be numerically demanding, and both Born-rule sampling and the sound velocity $\alpha$ require careful treatment.
5. Open research challenge: interacting fermions and non-Abelian topological order. A more ambitious direction is to characterize new transitions in monitored or decohered interacting-fermion systems and non-Abelian topological phases. Such phases are difficult to fabricate and verify in solid-state materials but may be more naturally realized as programmable quantum matter, where measurement and noise are intrinsic ingredients. Related work is ongoing in our group.

Acceptance / deliverables

• Documented code implementing Born-rule sampling and tensor-network contraction—or, where appropriate, Chalker–Coddington-type network evolution—for the Lyapunov exponents.
• Estimates of $c_{\rm eff}\equiv c_{\rm Casimir}$, with Monte Carlo error bars, for the Nishimori and weak self-dual points. The results should agree with the target values above and include a finite-size analysis over several circumferences $L$.
• A short report describing the model mappings, tensor-network or transfer-matrix construction, sampling and disorder averaging, and extracted central charges with uncertainties. Strong submissions should also study the full Lyapunov spectrum and/or the open metal–insulator transition of arXiv:2512.19786.

Background & references

• Conformal field theory
  • A. B. Zamolodchikov, JETP Lett. 43, 730 (1986) ($c$-theorem).
• Measurement-induced criticality
  • C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum circuits, arXiv:1908.08051.
  • A. Zabalo et al., Phys. Rev. Lett. 128, 050602 (2022), arXiv:2107.03393.
• Measurement-based preparation and Nishimori transitions
  • G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori's Cat: Stable Long-Range Entanglement from Finite-Depth Unitaries and Weak Measurements, arXiv:2208.11136.
  • E. H. Chen et al., Realizing the Nishimori transition across the error threshold for constant-depth quantum circuits, arXiv:2309.02863.
  • J. Y. Lee, W. Ji, Z. Bi, and M. P. A. Fisher, Decoding Measurement-Prepared Quantum Phases and Transitions: from Ising model to gauge theory, and beyond, arXiv:2208.11699.
• Topological memories and mixed-state criticality
  • E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002), arXiv:quant-ph/0110143.
  • A. Honecker, M. Picco, and P. Pujol, Nishimori point in the 2D $\pm J$ random-bond Ising model, arXiv:cond-mat/0010143.
  • F. Eckstein, B. Han, S. Trebst, and G.-Y. Zhu, Robust teleportation of a surface code and cascade of topological quantum phase transitions, arXiv:2403.04767.
  • M. Pütz, S. J. Garratt, H. Nishimori, S. Trebst, and G.-Y. Zhu, Learning transitions in classical Ising models and deformed toric codes, arXiv:2504.12385.
  • Q. Wang, R. Vasseur, S. Trebst, A. W. W. Ludwig, and G.-Y. Zhu, Decoherence-induced self-dual criticality in topological states of matter, arXiv:2502.14034 (Table 1 and Figs. 9(a) and 13: $c_{\rm Casimir}\equiv c_{\rm eff}$ and Lyapunov spectra).
  • G.-Y. Zhu, KITP chalk talk, KITP Program: Learning the Fine Structure of Quantum Dynamics in Programmable Quantum Matter (2025).
  • Z.-Q. Wan, X.-D. Dai, and G.-Y. Zhu, Revisiting Nishimori multicriticality through the lens of information measures, Phys. Rev. Research 8, 023059 (2026), arXiv:2511.02907.
  • F. Eckstein, B. Han, S. Trebst, and G.-Y. Zhu, Learning transitions of topological surface codes, arXiv:2512.19786.

No repository files, tests, or entry points are named. Start by reviewing the issue's transfer-matrix, Born-rule sampling, and Lyapunov-exponent requirements; done means documented code, finite-size analyses, and uncertainty-bearing central-charge estimates matching the Nishimori and weak self-dual targets.