Effective temperature in approximate quantum many-body states

In this work, we investigate a profound and previously unexplored aspect of approximate quantum states: the effective temperature that characterizes their spectral properties. The concept of temperature arises from statistical mechanics [5]. Here, we apply this concept to quantum many-body pure states, specifically to the approximate states obtained through optimizing different ansatzes. We investigate the effective temperature and its dynamics during variational training, using both energy expectation and fidelity as objectives.

Our work reveals a surprising universality in the effective temperatures of approximate quantum ground states, irrespective of the ansatz structure, objective function, training wellness, or underlying physical system. By decomposing the approximate states into the exact eigenstates of the system Hamiltonian, we observe a spectrum extending to the high-energy end with exponential decay of small decay factors, interpreted as inverse effective temperatures. The effective temperature also shows an intriguing two-stage behavior when approximating pure states of finite “temperature". The universal pattern could play an important role in understanding complex quantum systems or designing new classical and quantum algorithms. The effective temperature offers a new lens to assess the accuracy and reliability of different variational ansatzes and algorithms, potentially guiding the design of more efficient methods.

For an approximate state $|\psi\rangle$, we decompose it on the basis of the exact eigenstates $|\varepsilon_{i}\rangle$ of Hamiltonian $H$:

The spectral decomposition coefficients $|c_{i}|^{2}$ are the central focus of our study. We find that the pairs of $(\varepsilon_{i},|c_{i}|^{2})$ roughly follow an exponential decay for the approximate ground state, analogous with Boltzmann distribution,

where we denote the decay factor $\tilde{\beta}$ as the inverse effective temperature for the pure state $|\psi\rangle$ under investigation.

To gain deeper insights into the spectrum pattern of approximate quantum states, we evaluate additional target states beyond the ground state $|\varepsilon_{0}\rangle$. This set of states is directly parameterized with an inverse effective temperature $\beta$ as

where $Z(\beta)=\sqrt{\sum_{i}e^{-\beta\varepsilon_{i}}}$ is the normalization factor. These states can be understood as imaginary-time evolved states with time $\beta$ from the initial state $|\phi(0)\rangle=\frac{1}{Z(0)}\sum_{i}|\varepsilon_{i}\rangle$. Therefore, we call the target states imaginary-time evolved states (ITES). Exact ITES by default admits an exponential decay for spectral decomposition with inverse temperature $\beta$. We investigate how spectral properties change for approximate ITES $|\psi\rangle$ which are obtained by optimizing the fidelity objective $\mathcal{L}_{F}(\beta)=1-|\langle\phi(\beta)|\psi(\theta)\rangle|^{2}$. This objective reduces to the ground state target in the $\beta\rightarrow\infty$ limit. We find that the approximate ITES spectrum shows an evident two-stage behavior for all ansatzes: $\tilde{\beta}$ from approximate ITES successfully matches $\beta$ for small $\beta$ (high-temperature regime) and shows deviation $\tilde{\beta}<\beta$ for large $\beta$ (low-temperature regime). More importantly, the transition point $\beta^{*}$ characterizes the lowest effective temperature a given ansatz can reach and implies the intrinsic power of the corresponding ansatz. Notably, the conclusion remains qualitatively the same when the coefficient for each eigenstate in ITES acquires an extra random phase for generality, i.e. $|\phi(0)\rangle=\frac{1}{Z(0)}\sum_{i}e^{i\omega_{i}}|\varepsilon_{i}\rangle$.

In this work, we employ various quantum wavefunction ansatzes to demonstrate the universality of our findings, including (1) tensor-network based ansatz including matrix product states (MPS) [11, 12, 13] and projected entangled-pair states (PEPS) [14, 15], (2) neural quantum states (NQS) built on top of neural networks [16, 17], (3) output states from variational quantum circuits [18, 19] such as variational quantum eigensolvers (VQE), [20, 21] and (4) vector state ansatz (VEC) where each wavefunction component is directly modeled as a trainable parameter. We also comment on the relevance of our results to quantum approximate optimization algorithms (QAOA) [22]. We apply these ansatzes to different system Hamiltonians on different lattice geometries and optimize both objectives $\mathcal{L}_{E}$ and $\mathcal{L}_{F}$. The main findings of this work are summarized in Fig. 1.

We focus on the two-dimensional XXZ model on a square lattice with spin-$1/2$ degrees of freedom:

where $X(Y,Z)_{i}$ are Pauli matrices on lattice site $i$ and $\langle ij\rangle$ denotes the nearest neighbor sites. The phase diagram of this model, as given in Ref. [25], features $J_{z}^{c}=1$ that separates the spin-flipping phase and the antiferromagnetic phase when $J_{x}=J_{y}=1$. We impose periodic boundary conditions throughout this work. For $4\times 4$ lattice, we focus on the half-filling charge sector for spectral decomposition while the variational wavefunction ansatzes are still expressed and optimized in the full Hilbert space for simplicity. For $4\times 3$ lattice, we utilize the full spectral decomposition for most of the ansatzes while restricting to the half-filling sector for VQE states (see SM for details).

Using different ansatzes and variational optimization against $\mathcal{L}_{E}$ or $\mathcal{L}_{F}$, we obtain different approximate ground states. The typical spectral decomposition patterns $(\varepsilon_{i},|c_{i}|^{2})$ of these states are shown in Fig. 2 (a)(b). (See SM for more spectral decomposition results from other variational ansatzes.) We find a robust exponential relation in the spectral decomposition across different ansatzes and training stages. The exponential decay factor $\tilde{\beta}$ extracted from the fitting plays the role of inverse effective temperature. The training dynamics of $\tilde{\beta}$ for different ansatzes and objectives is shown in Fig. 3. We find that the effective temperature $1/\tilde{\beta}$ is high during training and this is particularly true for optimized approximate ground states where $\tilde{\beta}<0.3$ in most cases. Such a small $\tilde{\beta}$ contrasts sharply with imaginary-time evolution based approaches where a large $\tilde{\beta}$ is expected. During variational training, a general trend emerges where $\tilde{\beta}$ first increases and then decreases, implying an upper bound for the possible $\tilde{\beta}$ of a given ansatz. Besides, training dynamics of $\tilde{\beta}$ show similar patterns for both objectives within the same ansatz while $\tilde{\beta}$ for optimizing energy objectives $\mathcal{L}_{E}$ is typically higher due to the unequal penalty on different excited states. Conversely, fidelity objective equally penalizes all excited states and the finite $\tilde{\beta}$ in this case is attributed to the physical prior encoded in the ansatz structures. This understanding is validated by the near-zero $\tilde{\beta}$ for VEC ansatz, which contains the least physical prior in the structure. Furthermore, the $\tilde{\beta}$ trends are more closely aligned within the same family of ansatzes, reflecting its potential power to diagnose intrinsic properties in these ansatzes. The universality and validness of the results are also confirmed with other system sizes, models, and lattice geometries (see SM for details).

The results of $\tilde{\beta}$ of approximate ITES for different target $\beta$s are shown in Fig. 4 (a), where the two-stage behavior is evident. For each ansatz, there is a transition point $\beta^{*}$ when $\tilde{\beta}$ begins to deviate from $\beta$ and the associated spectrum pattern moves from Fig. 2 (c) to Fig. 2 (d). $\beta^{*}$ is a figure of merit as it marks the critical temperature that the ansatz can accurately represent and sets an upper bound on $\tilde{\beta}$ that an ansatz can reach. More importantly, $\beta^{*}$ is also correlated with the expressive capacity of the ansatz, as a higher $\beta^{*}$ generally implies a better fidelity. In the low temperature limit $\beta\rightarrow\infty$, the results for approximate ITES are reduced to the results for approximate ground states by optimizing $\mathcal{L}_{f}$.

We further remark on the fidelity results shown in Fig. 4 (b) with varying methods and $\beta$s. For low-accuracy ansatzes, the fidelity gets improved with increasing $\beta$ as $|\phi(\beta)\rangle$ of larger $\beta$ is closer to the ground state and has smaller entanglement. Such less entangled states can be better suited in ansatzes of limited expressiveness. Conversely, for high-accuracy ansatzes such as tensor network ansatz with large bond dimensions or VEC, the expressive capacity is sufficient for targeting any ITES, and the bottleneck is the optimization difficulty. Since the number of optimization steps is fixed in our simulation, a slower optimization speed results in a worse final fidelity with increasing $\beta$. In other words, when the target is closer to the ground state manifold, the optimization landscape is harder to navigate (see quantitative analysis for the expressiveness and optimization hardness in the SM).

The effective temperature has profound implications for understanding numerical methods and many-body systems. One practical implication of such a high-temperature tail in the spectrum is mentioned in Ref. [28], where sampling based energy variance estimation becomes more fragile than expected. The variance $H^{2}$ is just an example for operator $f(H)$, with expectation $\langle\psi|f(H)|\psi\rangle=\sum_{i}|c_{i}|^{2}f(\varepsilon_{i})$. The operator expectation is highly sensitive to small high-energy components when $f(\varepsilon)$ is significantly larger for high-energy inputs. The effective temperature captures finer details than fidelity as demonstrated in Fig. 4: while the fidelity remains the same for larger $\beta$, the effective temperature continues to vary. This quantity also has strong relevance to the expressive capacity of ansatzes as $\beta^{*}$ values differ for different ansatzes, which further confirms that the two-stage behavior in approximating ITES is not a numerical precision issue. Consequently, the metric and the methodology explored in this work serve as a guiding principle to help design better approximate quantum ansatzes and variational algorithms.

The effective temperature metric opens up several intriguing future research directions. We can quantitatively investigate the scaling of effective temperature $\tilde{\beta}$ and its critical value $\tilde{\beta}^{*}$ with respect to varying system sizes, Hamiltonian parameters, and ansatz structure parameters, to gain deeper insights into the physics of the system and the intrinsic patterns in numerical methods. It is also interesting to explore whether some approximation methods can be developed to estimate the effective temperature beyond exact diagonalization sizes. The ITES proposed in this work are also of independent academic interest, as they provide a platform to explore the trade-off between expressiveness and optimization hardness, as well as host a putative phase transition at $\beta^{*}$, the nature and universal behavior of which merit future exploration. It is also crucial to validate the general picture of spectrum patterns on more ansatzes including mean-field ansatz, physics-inspired ansatz [29] and advanced hybrid variational ansatzes [30, 31], and more models including integrable systems, many-body localized systems and fermionic systems.

In this work, we have identified a universal pattern in the spectrum of approximate ground states. We define the effective temperature $\tilde{\beta}$ to characterize the exponential decay in the spectrum and investigate the behavior of $\tilde{\beta}$ with different systems, ansatzes, objectives, and training steps. To better understand the underlying mechanism for this metric, we further propose ITES targets and identify the two-stage behaviors in approximating ITES with the figure of merit critical inverse effective temperature $\beta^{*}$. We believe that the universal picture presented here provides a fresh and powerful perspective on benchmarking numerical algorithms and understanding quantum many-body systems.

Matrix product states. We use MPS with periodic boundary conditions. The amplitude for the ansatz is given as

where $i_{j}=0$ or $1$ represents the computational basis and $A^{(j)}$ are $L$ different tensors with shape $(\chi,\chi,2)$. $\chi$ is the bond dimension controlling the expressiveness of the MPS ansatz. For $\chi=2^{L/2}$, the ansatz can be an exact representation for any wavefunction in principle even with open boundary conditions. The tensor network graphic representation for the ansatz in Eq. (7) is shown in Fig. 5. Note that we don’t explicitly impose the normalization conditions by canonicalization but leave the MPS ansatz unnormalized. The Hamiltonian expectation is evaluated as

where normalization is explicitly accounted for. For 2D systems, we use the natural order from site $1$ to $L$ to encode the system sites into 1D MPS. Since we need to consider the full spectrum properties in the calculation, the system size accessible is limited to $L=16$. Therefore, we don’t rely on the density matrix renormalization group optimization algorithm. Instead, we directly obtain the full wavefunction by contracting the MPS according to Eq. (7) and directly evaluate the Hamiltonian expectation by sparse matrix-vector multiplication where the objective can then be differentiated and variational optimized via gradient descent. For variational optimization, each element in $A^{(j)}$ is randomly initialized according to the Gaussian distribution of mean $0$ and standard deviation $0.1$. The total number of variational parameters for MPS ansatz is $2L\chi^{2}$.

Projected entangled-pair states. The tensor network graphic representation of the PEPS ansatz with periodic boundary conditions is shown in Fig. 6. Similarly, the ansatz is unnormalized by default. During optimization, $L=L_{x}\times L_{y}$ tensors with shape $(\chi,\chi,\chi,\chi,2)$ are randomly initialized with each element following Gaussian distribution of mean $0$ and standard deviation $0.1$. Henceforth, the total number of variational parameters for PEPS is $2L\chi^{4}$. In the numerical calculation, we first contract the entire PEPS tensor network with physical legs open to directly obtain the full unnormalized wavefunction since the system size is small. We can then compute fidelity or Hamiltonian expectation directly and integrate the whole computation graph with automatic differentiation and variational optimization.

Neural quantum states. For NQS, the probability amplitude is given by the output of neural networks with the input the computational basis, i.e.

where $f$ corresponds to the neural network. In this work, we employ a deep neural network with the architecture shown in Fig. 7. Given the small system size, we do not run the variational Monte Carlo method with sampling to optimize the NQS. Instead, similar to the MPS and PEPS cases, we directly evaluate the neural network output on all $2^{L}$ computational basis inputs to first obtain the (unnormalized) full wavefunction and optimize infidelity or energy based on the wavefunction. The total number of trainable parameters is of order $O(LWD)$, where $D$ is the number of repetitions of the ResBlock and $W$ is the width of intermediate fully connected layers. The network width $W$ controls the expressiveness of the NQS ansatz.

Variational quantum eigensolver states. In VQE methods, we use the output quantum states from the parameterized quantum circuits (PQC) and the trainable parameters correspond to rotation angles of quantum gates on the PQC. The PQC $U=U_{v}U_{i}$ consists of an initialization block $U_{i}$ and $d$ variational blocks $U_{v}=\prod_{i}^{d}U_{vi}$. The initialization block prepares the Bell pair $1/\sqrt{2}(|01\rangle-|10\rangle)$ on nearest neighbor qubits so that the initial state is in the $J_{tot}=0$ sector. Each variation block is given by: $U_{vi}=\prod_{j}^{L}e^{-i\theta^{(0)}_{j}Z_{j}}e^{-i\theta^{(1)}_{j}Y_{j}}\prod_{\langle jk\rangle}e^{-i\theta^{(2)}_{jk}\text{SWAP}_{jk}}$, where the total number of circuit parameters $\theta$ is of the order $O(Ld)$ and $\text{SWAP}_{jk}=1/2(X_{j}X_{k}+Y_{j}Y_{k}+Z_{j}Z_{k}+I_{j}I_{k})$ is the swap gate on qubits $j$ and $k$. These parameterized swap gates are applied on the bond between nearest neighbor sites according to the lattice geometry. For 2D square lattice, these two-qubit gates are first applied on all rows following all columns without periodic bonds across the opposite boundaries. For 1D lattice, these two-qubit gates are applied on the system in a ladder fashion with a final periodic bond gate, i.e. the entangling gates apply on the qubit pairs $(1,2),(2,3),\cdots(L-1,L),(L,1)$. A schematic of the PQC structure investigated in this work is shown in Fig. 8. The approximate state we use is $|\psi\rangle=U|0^{L}\rangle$ accordingly. Our numerical simulation for expectation value and fidelity omits quantum noise error and shot sampling error inevitable on real quantum devices. The circuit parameter gradients are obtained via automatic differentiation in our simulation and can be obtained via parameter shift [32] on quantum processors.

Vector states. Since the system size under investigation is small, we can directly build up the so-called vector states ansatz where the wavefunction amplitude of each basis is directly modeled as a trainable parameter. For a spin-1/2 system of size $L$, $2^{L}$ complex parameters are required, namely, we treat $\langle i_{1}i_{2}\cdots i_{L}|\psi\rangle$ as variational parameters. In our work, the real part and imaginary part of these complex parameters are independently initialized (Gaussian distribution of mean $0$ and standard deviation $0.1$) and optimized. This is because the Adam optimizer does not work as expected with complex training parameters. This ansatz is expected to encode the least physical prior and be the most unbiased.

Supplemental Material for “Effective Temperature in Approximate Quantum Many-body States”