Stochastic Maximum Likelihood in Quantum Tomography

• Stochastic Maximum Likelihood (SML) is an estimation principle that maximizes the expected likelihood under uncertainty, particularly useful in quantum state tomography.
• It employs stochastic mirror descent with Burg entropy to maintain full-rank iterates while managing high-dimensional optimization problems efficiently.
• Empirical evaluations demonstrate that SML achieves rigorous non-asymptotic convergence rates and outperforms traditional methods in computational scalability.

Stochastic Maximum Likelihood (SML) is an estimation principle applied across diverse domains where statistical models and their likelihood functions are subject to randomness due to noise, latent variables, or stochastic transitions. Central to SML is the maximization of the expected likelihood—often only approximately computable—under uncertainty, with efficiency and scalability achieved via stochastic approximation techniques. In contemporary quantum state tomography, SML is particularly crucial, given the exponential growth of data and parameter space as system dimensionality increases.

1. Definition and Context of SML in Quantum State Tomography

SML concerns the estimation of a quantum state's density matrix $\rho$ from measurement outcomes $A_i$, where both the number of observations ($n$) and state dimension ($d$) scale rapidly with system size. The maximum-likelihood quantum state estimation problem is formalized as the convex optimization:

$$\hat{\rho} \in \arg \min_{\rho \in \mathcal{D}} f(\rho), \quad f(\rho) = \frac{1}{n} \sum_{i=1}^n -\log \mathrm{tr}(A_i \rho)$$

with constraints

$$\mathcal{D} = \{\rho \in \mathbb{C}^{d \times d} : \rho = \rho^*,\; \rho \geq 0,\; \mathrm{tr}(\rho) = 1\}$$

Here, $A_i$ are Hermitian positive semi-definite measurement matrices, and $\rho$ is a density matrix (positive semidefinite, unit trace). SML approaches are essential for tractable estimation when $n$ and $d$ are large.

2. Stochastic Mirror Descent with Burg Entropy

To address the scalability constraints inherent in quantum tomography, the paper proposes a first-order SML algorithm based on stochastic mirror descent (SMD) with Burg entropy as the mirror map, leveraging a sequence of iterates that maintain full rank.

Iterative Scheme

Each iteration $t$ executes the following sequence:

1. Averaging: Compute the running average of previous iterates:

$$\overline{\rho}_t = \frac{1}{t} \sum_{\tau=1}^t \rho_\tau$$

2. Stochastic Gradient Evaluation: Uniformly randomly select $i_t \in \{1,...,n\}$ and calculate

$$\nabla f_t(\overline{\rho}_t) = \frac{ -A_{i_t} }{ \mathrm{tr}(A_{i_t} \overline{\rho}_t) }$$

3. Mirror Descent Update: Update

$$\rho_{t+1} = \arg \min_{\rho \in \mathcal{D}} \Big[ \eta\, \mathrm{tr}\left( \nabla f_t(\overline{\rho}_t)(\rho - \rho_t) \right) + D_h(\rho, \rho_t) \Big]$$

where $\eta$ is the step size and $D_h(\rho, \rho_t)$ is the Bregman divergence induced by Burg entropy $h(\rho) = -\log\det(\rho)$:

$$D_h(\rho, \sigma) = h(\rho) - h(\sigma) - \mathrm{tr} \left( \nabla h(\sigma)(\rho - \sigma)\right)$$

with $\nabla h(\rho) = -\rho^{-1}$.

The key computational advantage arises from mirror descent—updates require a single matrix eigendecomposition and a projection onto the simplex, compared to multiple expensive computations typical in alternate methods.

3. Computational Complexity and Scalability

The per-iteration computational complexity is dominated by the eigendecomposition:

$O(d^3) \quad \text{per iteration}$

This is independent of sample size $n$, making it superior for large-$n$ settings. In contrast, standard projected gradient descent incurs $O(d^3 + n d^\omega)$ time per iteration, with dependence on expensive tensor or matrix operations ($\omega > 2$ typically refers to the matrix multiplication exponent). Alternative stochastic methods such as Q-Soft-Bayes require more expensive matrix logarithms and exponentials per iteration.

4. Convergence Rate and Statistical Guarantees

The algorithm achieves a non-asymptotic convergence rate:

$$\mathbb{E} \left[ f(\overline{\rho}_T) - \min_{\rho \in \mathcal{D}} f(\rho) \right] \leq O\left( \sqrt{ \frac{d \log T}{T} } \right)$$

This guarantees that the expected optimization error vanishes at the stated rate in $T$ (number of iterations) and $d$ (matrix dimension), as formalized in Theorem 1 of (Tsai et al., 2022).

Importantly, the use of Burg entropy ensures all iterates are full rank, preventing stalling due to zero eigenvalues that hinder projected gradient descent in quantum tomography.

5. Empirical Performance and Comparison

Experiments indicate robust performance and dramatic speedup over previous stochastic first-order methods (e.g., 2.3x faster than Stochastic Q-Soft-Bayes in elapsed time for large $d$). The algorithm demonstrates scalability up to six-qubit systems ($d=64$, $n=409400$) and outpaces previous approaches in scenarios where both $d$ and $n$ are large.

While some non-stochastic algorithms can be faster in specific instances, they lack rigorous convergence guarantees provided by the proposed SML method.

6. Implementation Considerations and Limitations

Implementation efforts should focus on efficient eigendecomposition routines and careful selection of step size $\eta$. The method is insensitive to the number of measurement outcomes, making it robust for high-throughput quantum experiments. Nevertheless, performance in extremely large $d$ regimes will ultimately be bounded by available computational resources (RAM, CPU/GPU speed for linear algebra operations).

Deployment for practical quantum state reconstruction systems should leverage optimized numerical libraries for eigendecomposition and simplex projection. The algorithm’s independence from $n$ naturally allows for distributed settings where measurement data is abundant.

7. Application Scope

The SML mirror descent approach with Burg entropy is tailored to quantum state tomography but conceptually generalizes to other domains featuring full-rank density matrix estimation problems under log-likelihood objectives. Its probabilistic guarantee, computational independence from sample size, and explicit management of high-dimension update steps render it well-suited to other matrix-based statistical estimation settings where stochastic approximation and convexity are exploitable.

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References: Key formulations, numerical results, and comparisons referenced from "Faster Stochastic First-Order Method for Maximum-Likelihood Quantum State Tomography" (Tsai et al., 2022).