Thermal Pure Quantum States (TPQ)

• TPQ states are pure quantum states that, in the thermodynamic limit, reproduce all equilibrium properties of statistical ensembles.
• They utilize numerical and quantum algorithms such as Lanczos, Chebyshev expansion, and QITE to obtain observables with exponentially suppressed errors.
• TPQ methods are pivotal for simulating complex quantum materials, gauge theories, and real-time dynamics, offering both computational and conceptual advantages.

A thermal pure quantum (TPQ) state is a single pure quantum state that, with overwhelming probability in the thermodynamic limit, reproduces all equilibrium properties of a statistical ensemble (canonical or microcanonical) for quantum many-body systems. TPQ states provide a pure-state alternative to mixed-state (density operator) prescriptions for quantum statistical mechanics, allowing exact or highly accurate evaluation of thermodynamic observables, correlation functions, and response properties with computational and conceptual advantages in both classical and quantum simulation frameworks. The construction and rigorous properties of TPQ states have enabled new methodologies in tensor networks, quantum algorithms, and the simulation of highly entangled quantum materials and gauge theories.

1. Definition and Mathematical Structure of TPQ States

Let $H$ denote the Hamiltonian of a quantum system with Hilbert-space dimension $D$. The canonical TPQ state at inverse temperature $\beta=1/T$ is defined by

$$|\beta\rangle = \frac{e^{-\frac{\beta}{2}H} |r\rangle}{\|\,e^{-\frac{\beta}{2}H} |r\rangle\|},$$

where $|r\rangle$ is a normalized random vector in the Hilbert space, typically taken to have components that are independent complex Gaussian or Haar-random with mean zero and variance $1/D$ in any orthonormal basis. For any “mechanical" or local observable $O$, in the thermodynamic limit,

$$\langle\beta|O|\beta\rangle \approx \frac{\mathrm{Tr}[e^{-\beta H}O]}{\mathrm{Tr}\,e^{-\beta H}},$$

with deviations vanishing exponentially in system size (Sugiura et al., 2013, Sugiura et al., 2013, Gohlke et al., 2023).

Microcanonical TPQ states are constructed via iterative application of a shifted Hamiltonian operator to $|r\rangle$, $(l - h)^k |r\rangle$, where $h = H/N$, $l > \max \mathrm{Spec}(h)$, yielding states sharply peaked at a target energy density. Grand-canonical TPQ (gTPQ) states generalize this to fluctuating particle number by filtering with $e^{-\frac{\beta}{2}(H-\mu N)}$ (Hyuga et al., 2014).

In all cases, a single TPQ state suffices: expectation values and norms yield ensemble-averaged observables and thermodynamic quantities with errors $O(e^{-cN})$ for $N$ sites.

2. Rigorous Typicality, Fluctuations, and Error Bounds

TPQ states are founded on canonical typicality and large deviation principles. The key result states that for any local or mechanical observable $A$,

$$\Pr\left(|\langle A\rangle_{\mathrm{TPQ}} - \langle A\rangle_{\mathrm{ens}}| \geq \epsilon\right) \leq \frac{1}{\epsilon^2} \frac{\langle(\Delta A)^2\rangle_{2\beta}}{e^{2N\beta(f(\frac{1}{2\beta})-f(\frac{1}{\beta}))}},$$

where $f(\beta)$ is the free energy density and the numerator is at most $O(N^{2m})$, $m$ degree of $A$ (Sugiura et al., 2013, Sugiura et al., 2013).

For gTPQ, similar exponential concentration in volume $V$ is established for both the norm and observable fluctuations: $$P\left(|\langle O\rangle_{\mathrm{TPQ}} - \langle O\rangle_{\mathrm{ens}}| \geq \epsilon\right) \leq D_V^2 / \epsilon^2,$$ with $D_V^2 = O(V^{2m}) e^{-\Theta(V)}$ (Hyuga et al., 2014).

Thus a single realization yields thermodynamic and local observables to arbitrary accuracy as $N\to\infty$.

3. Construction Algorithms: Numerical and Quantum Implementations

Classical construction of TPQ states proceeds by (i) generating a random vector $|r\rangle$, (ii) acting with $e^{-{\beta}H/2}$ via Krylov (Lanczos or Chebyshev), Taylor expansion, or power iteration, and (iii) normalizing and measuring observables (He et al., 22 Oct 2025, Iwaki et al., 2020).

Efficient algorithms for imaginary-time evolution, including dynamic scaling-and-Taylor and block-encoding/QSVT techniques, have led to significant speed-ups and numerical stability, enabling TPQ simulations on $4\times 4$ Hubbard clusters over a wide range of $\beta$ with errors $<10^{-3}$ (He et al., 22 Oct 2025). Numerically, performance and scaling are benchmarked against ED, Lanczos, and QMC, with practical advantages at moderate to large $N$.

Quantum algorithms implement TPQ preparation via (i) Haar-random state initialization (using random circuits or Clifford $k$-designs), (ii) quantum imaginary time evolution (QITE), polynomial filtering via eigenvalue transformation, or amplitude amplification, and (iii) measurement—either projective (for static observables) or via interferometric protocols (for correlation functions) (Powers et al., 2021, Davoudi et al., 2022, Mizukami et al., 2022, Watts et al., 2023, Seki et al., 2022).

For gauge theories, modifications incorporate penalty Hamiltonians to enforce local constraints, yielding “physical TPQ” states that remain within gauge-invariant sectors (Davoudi et al., 2022, Davoudi et al., 2022).

4. Tensor Network Representations and Volume-Law Entanglement

Standard matrix product states (MPS) are efficient for states with area-law entanglement, but canonical TPQ states exhibit volume-law entanglement: the von Neumann entropy $S(\ell)$ on block length $\ell$ grows as $S(\ell)\sim \ell\, s(\beta)$, with $s(\beta)$ the thermal entropy density (Iwaki et al., 2020, Yoneta, 19 Jul 2024).

To efficiently represent TPQ states with tensor networks, several strategies have been developed:

• TPQ-MPS Construction: By attaching auxiliary chains (“entanglement baths”) to the ends of the MPS, the bond dimension $\chi$ accommodates volume-law entanglement up to $\sim2\ln\chi$ in the bulk (Iwaki et al., 2020). This approach reproduces thermal observables with $\chi\sim20$–$40$ for 1D and small 2D systems (Gohlke et al., 2023).
• Antiunitary Symmetry and EAP States: In systems with antiunitary symmetries (e.g., time-reversal, complex conjugation), structured "entangled antipodal pair" reference states allow for deterministic TPQ construction and explicit mappings to low-bond-dimension MPS or PEPS, enabling tractable simulation up to $N\approx 200$ spins in 1D and $N\approx 60$ in 2D (Yoneta, 19 Jul 2024).
• Truncation Procedures: During each imaginary-time step, singular-value truncation and bond dimension controls retain dominant entanglement, efficiently discarding high-energy states as $\beta$ increases (Gohlke et al., 2023).

These methods combine TPQ typicality with tensor network scalability, yielding pure-state descriptions of thermal equilibrium even in highly entangled, topologically ordered phases.

5. Thermodynamic Observables and Real-Time Dynamics

Once a TPQ state is prepared, the full suite of equilibrium properties is accessible:

• Expectation Values: $$\langle O\rangle_\beta = \langle\beta|O|\beta\rangle$$ reproduces ensemble averages.
• Free Energy and Entropy: $$F(\beta)\approx -\frac{1}{\beta}\ln\langle\Psi_\beta|\Psi_\beta\rangle$$; entropy follows as $S(\beta) = \beta(E-F)$.
• Specific Heat: $C(\beta)=dE/dT$ is computed from energy derivatives.
• Correlation Functions: TPQ states reproduce multi-point correlations with errors vanishing exponentially in $N$.
• Non-Equilibrium and Response: Time evolution of TPQ states under quantum quenches gives access to both linear ($\chi^{(1)}(\omega)$) and nonlinear susceptibilities at arbitrary field strength, with rigorous error control (Endo et al., 2018). Intrinsic “thermalization” dynamics and rich excitation spectra are captured by tracking observable transients.

In quantum algorithms, real-time correlation functions are extracted via Ramsey/interferometry circuits, and effective thermalization of local subsystems can be directly observed experimentally (Mitchison et al., 2021).

6. Applications: Quantum Materials, Gauge Theories, and Beyond

TPQ methods have been applied to strongly correlated quantum lattices, frustrated antiferromagnets, spin liquids, Fermi-Hubbard and Bose-Hubbard models, and finite-density gauge theories:

• Kitaev Honeycomb Model: TPQ-MPS on cylinder topology faithfully tracks the transition from high-temperature paramagnet to quantum spin liquid ground state, capturing Majorana physics and specific-heat features at $T_H/K\approx 0.5$ and $T_L/K\approx 0.016$. The approach drastically outperforms exact diagonalization and MPDO-based thermal simulations in accessible temperature range and system size (Gohlke et al., 2023).
• Gauge Theories: TPQ formalism with gauge constraint penalties produces phase diagrams for $1+1$D $\mathbb{Z}_2$ gauge theory, recapitulating chiral phase transitions and capturing real-time correlator dynamics. Quantum circuits using QITE and random-state preparation are shown to be robust to noise and gate errors, with resource requirements compatible with near-term hardware (Davoudi et al., 2022, Davoudi et al., 2022).
• Quantum Algorithms for SDP: TPQ-based quantum SDP solvers approach the optimal quadratic speedup of Gibbs-sampler-based methods with substantially reduced ancilla overhead and additional error vanishing exponentially with system size under spectral conditions (Watts et al., 2023).
• Hydrodynamics: Local TPQ (“$\ell$TPQ”) states enable pure-state descriptions of local thermal equilibrium and emergent quantum hydrodynamics, supporting the second law and fluctuation relations with exponentially suppressed error for sufficiently large fluid cells (Tsutsui et al., 2021).

Quantum and classical benchmarking studies confirm exponential convergence to ensemble accuracy ($<1\%$ error) down to low temperatures and across doping and interaction regimes, with statistical uncertainty often outperforming conventional QMC or Lanczos methods (Roy et al., 3 Jul 2025, He et al., 22 Oct 2025, Hyuga et al., 2014).

7. Limitations and Open Questions

While TPQ states are universal in their applicability at finite temperature for gapped, short-range systems, certain caveats remain:

• Phase Transitions: Near criticality, the standard large-deviation bounds require refinement, and convergence can be slower due to critical fluctuations (Sugiura et al., 2013).
• Entanglement and Tensor Network Cost: Representation of volume-law TPQ states as MPS or PEPS still necessitates exponential resources for very large systems or in higher dimensions unless auxiliary chain methods or symmetry-based mappings are exploited (Yoneta, 19 Jul 2024).
• Quantum Algorithm Scalability: Quantum TPQ algorithms achieve $O(\sqrt{D})$ scaling in state preparation complexity (where $D$ is the Hilbert-space size), a quadratic improvement over classical, but still exponential in system size. Nonetheless, for problems where classical methods are hampered by sign problems or sampling cost (e.g., frustrated magnets, doped gauge theories), quantum TPQ approaches offer unique advantages (Mizukami et al., 2022, Watts et al., 2023).
• Generalizations and Extensions: Current research seeks to extend TPQ methods to relativistic quantum fields, integrable systems with generalized hydrodynamics, and settings with strong disorder.

TPQ states represent a fundamental and practical advance in quantum statistical mechanics, bridging microscopic quantum entanglement, thermodynamics, and efficient simulation across platforms (Sugiura et al., 2013, Sugiura et al., 2013, Hyuga et al., 2014, Gohlke et al., 2023, He et al., 22 Oct 2025, Iwaki et al., 2020, Yoneta, 19 Jul 2024).