Randomized truncation of quantum states

TL;DR

The paper addresses the problem of approximating a pure quantum state by mixtures of low-coherence, $k$-sparse or Schmidt-$k$ states. It develops a max-entropy sampling framework and convex-optimization-based algorithms to compute optimal randomized truncations for both trace distance and robustness, with efficient sampling of the resulting ensembles. The key contributions include polynomial-time methods to obtain the optimal trace-distance and robustness values and sampling procedures, a detailed analysis of the associated ensembles, and explicit connections to the $k$-top and $k$-support norms. The results yield quadratic improvements in many cases over deterministic truncation, enable memory-efficient Monte Carlo-like truncations for tensor networks, and are demonstrated numerically on matrix product state truncations, suggesting practical impact for quantum-state simulations. The work also clarifies computational hardness for general mixed-state approximations and situates the approach within a broader ecosystem of related randomized algorithms and resource-theoretic perspectives.

Abstract

A fundamental task in quantum information is to approximate a pure quantum state in terms of sparse states or, for a bipartite system, states of bounded Schmidt rank. The optimal deterministic approximation in each case is straightforward, and maximizes the fidelity: keep the largest entries or singular values. On the other hand, random mixtures of sparse states can achieve quadratically improved trace distances, and yield nontrivial bounds on other distance measures like the robustness. In this work, we give efficient algorithms for finding mixtures of sparse states that optimally approximate a given pure state in either trace distance or robustness. These algorithms also yield descriptions of efficiently samplable ensembles of sparse, or less-entangled, states that correspond to these optimal mixed approximations. This can be used for the truncation step of algorithms for matrix product states, improving their accuracy while using no extra memory, and we demonstrate this improvement numerically. Our proofs use basic facts about convex optimization and zero-sum games, as well as rigorous guarantees for computing maximum-entropy distributions.

Figures (5)

• Figure 1: With $d=35$, $k=15$, and darker cells representing larger values: (a) Quantum state given by a random unit vector $v\in\mathbb{R}^d$ with positive entries sorted in decreasing order. (b) Approximation obtained by keeping the first $k$ entries of $v$ and normalizing, which yields $\widetilde{v}_{1:k}$. (c) Optimal approximation $\sigma^\star$ of $vv^\top$, with respect to trace distance, by a convex combination of states given by $k$-sparse unit vectors.
• Figure 2: Illustration of the geometric intuition underlying cases where randomized truncation offers a significant improvement in approximating a pure state $vv^\top$.
• Figure 3: With $d=200$ and $k=100$: (a) Example of a positive unit vector $v\in \mathbb{R}^d$ and the corresponding optimal $m\in\mathbb{R}^d$ derived in \ref{['lem:m_restricted_form']}. This $m$ has all three regions described in \ref{['eq:tilde_m_defn']}, but for other choices of $v$ and $k$, one might see only one or two of these regions. Vertical dotted lines are at $k-r$ and $\ell-1$, where in this case $r=39$ and $\ell=157$. (b) Marginal probability of drawing a $k$-sparse vector with a nonzero weight on the $i^\text{th}$ index, as a function of $i$, with the optimal ensemble of states described in \ref{['sec:optimal_density_matrix_td']}. In this case, the first $60$ entries are kept deterministically.
• Figure 4: Plot of the advantage $\textnormal{adv}(\gamma)$ for power law states $v$ with parameter $\gamma$ using randomized truncation, in the limit as $d\to\infty$ and holding $\varepsilon=1-F_k(v)^2$ fixed. Here, advantage is defined as the exponent of the leading-order contribution to the optimal trace distance, as a function of $\varepsilon$, and therefore lies between $1/2$ and $1$. In other words $\text{adv}(\gamma) = \lim_{\varepsilon\rightarrow 0}\lim_{d\rightarrow \infty} \log(T_k(v))/\log(\varepsilon)$, where $v$ is defined in \ref{['eq:power_law_entries']}. The true value of $\textnormal{adv}(\gamma)$ on the interval $(1/2,1)$ is in the shaded region.
• Figure 5: Estimating $\langle Z_5\rangle$ (the Pauli-Z operator on the 5$^\text{th}$ qubit) with respect to four randomly generated states on 9 qubits. We first express each state as an MPS in canonical form. We then replace the Schmidt coefficient tensors along each of the 8 edges with a diagonal tensor whose entries obey a power law scaling with parameter $\gamma$, as in \ref{['eq:power_law_entries']}, and renormalize. The estimates are obtained using an ensemble of matrix product states at a bond dimension cutoff. The dtrunc estimate is obtained from the standard, deterministic truncation of the bonds. The rtrunc (TD) estimate is the sample mean of 100 samples obtained by applying the optimal (bipartite) randomized truncation in trace distance to each bond. For smaller values of $\gamma$, there is a clear improvement using randomized truncation.

Theorems & Definitions (53)

• Theorem 1.1: Informal
• Definition 2.1: $k$-sparse mixture, $\mathcal{I}_k$
• Definition 2.2: Schmidt number $k$, $\mathcal{S}_k$
• Definition 2.3: Fidelity with $\mathcal{I}_k$ or $\mathcal{S}_k$
• Lemma 2.4
• Lemma 2.5
• proof
• Definition 2.6: Top-$k$ norm
• Definition 2.7: Trace distance with $\mathcal{I}_k$ or $\mathcal{S}_k$
• Definition 2.8: Robustness with $\mathcal{I}_k$ or $\mathcal{S}_k$