finite-temp-quantum-krylov

name	finite-temp-quantum-krylov
description	Finite-temperature quantum Krylov method for computing thermal properties of quantum many-body systems from real-time overlaps. Use when analyzing quantum many-body systems at finite temperatures, computing thermal observables, or avoiding thermal state preparation in quantum simulations.

Finite-Temperature Quantum Krylov Method

Method for evaluating finite-temperature properties of quantum many-body systems without requiring thermal-state preparation at each target temperature.

Core Concept

Traditional quantum approaches require thermal-state preparation at each target temperature, making low-temperature calculations demanding in terms of circuit depth and computational cost. This method uses real-time overlaps to extract finite-temperature properties.

Activation Keywords

• finite-temperature quantum
• quantum Krylov method
• thermal properties quantum
• quantum many-body temperature
• real-time overlaps
• thermal state preparation alternative

Mathematical Framework

Quantum Krylov Subspace

The method constructs a Krylov subspace from real-time evolution:

|ψ(t)⟩ = e^(-iHt)|ψ₀⟩

Where:

• H is the Hamiltonian
• |ψ₀⟩ is an initial state (typically the ground state or a simple product state)
• t is real time

Finite-Temperature Observables

Thermal expectation values are computed via:

⟨O⟩_β = Tr[O e^(-βH)] / Tr[e^(-βH)]

Where β = 1/(k_B T) is the inverse temperature.

Real-Time Overlap Approach

Instead of preparing thermal states, the method:

1. Evolves initial states in real time
2. Computes overlaps between time-evolved states
3. Extracts thermal information from these overlaps using analytical continuation or spectral methods

Implementation Steps

Step 1: State Preparation

Prepare an initial state |ψ₀⟩:

• Ground state (for low-temperature physics)
• Random product state (for high-temperature regime)
• Symmetry-preserving initial state

Step 2: Real-Time Evolution

Evolve the state for discrete time steps:

# Pseudocode
for t in time_grid:
    |ψ(t)⟩ = time_evolve(H, |ψ₀⟩, t)
    overlaps[t] = compute_overlap(|ψ(t)⟩, |ψ₀⟩)

Step 3: Overlap Collection

Collect real-time overlap data:

• ⟨ψ₀|ψ(t)⟩ (survival probability)
• ⟨ψ(t)|O|ψ(t)⟩ (time-dependent observables)
• Connected correlation functions

Step 4: Thermal Extraction

Extract finite-temperature properties:

1. Spectral Function Reconstruction

   • Use maximum entropy or Prony methods
   • Extract density of states from overlap decay

2. Partition Function Estimation

   Z(β) ≈ Σ_n |⟨ψ₀|n⟩|² e^(-βE_n)
   

3. Observable Computation

   ⟨O⟩_β = Σ_n ⟨n|O|n⟩ |⟨ψ₀|n⟩|² e^(-βE_n) / Z(β)
   

Advantages

1. No Thermal State Preparation: Avoids costly imaginary-time evolution
2. Single Real-Time Evolution: One time evolution yields all temperatures
3. Lower Circuit Depth: Real-time evolution often shallower than imaginary-time
4. Continuous Temperature Access: Extract any temperature from same data

Limitations

1. Analytic Continuation: Requires careful handling of ill-posed inversion
2. Signal-to-Noise: Long-time overlaps may have poor signal quality
3. Initial State Dependence: Results depend on choice of |ψ₀⟩
4. Spectral Resolution: Limited by maximum evolution time

Applications

• Condensed Matter Physics: Study quantum phase transitions at finite T
• Quantum Chemistry: Finite-temperature properties of molecules
• Quantum Magnetism: Thermal properties of spin systems
• High-Energy Physics: Finite-temperature field theories

Tools Used

• exec: Run quantum simulation code (Qiskit, Cirq, PennyLane)
• python: Numerical analysis of overlaps, analytic continuation
• write: Save computed thermal properties, analysis results

References

• arXiv:2604.10543v1 (2026) - "Finite-temperature quantum Krylov method from real-time overlaps"
• Quantum Krylov methods literature
• Maximum entropy methods for spectral reconstruction

Related Skills

• quantum-simulation: General quantum simulation techniques
• quantum-many-body: Quantum many-body physics methods
• thermal-quantum-states: Thermal state preparation and manipulation