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Bypassing orthogonalization in the quantum DPP sampler

Michaël Fanuel, Rémi Bardenet

TL;DR

This work targets exact sampling of projection DPPs on a quantum computer by removing the traditional $\mathcal{O}(nr^2)$ QR preprocessing bottleneck. It introduces column normalization to form $\mathsf{X}$ and uses Clifford loaders to load the data, coupled with a rejection sampler whose acceptance is $a = \det(\mathsf{X}^\top \mathsf{X})$; amplitude amplification and a sketch-based estimate of $a$ further reduce the number of rejections and overall cost. A key technical contribution is a novel DPP with a nonsymmetric (skew-symmetric) correlation kernel, exemplified by determinantal measures over dimer-rooted forests and connections to Pfaffians, which arises naturally in the graph-based instantiations and through the loading scheme. The numerical and implementation results in Qiskit demonstrate feasibility and offer a path toward practical quantum-speedups for sampling projection DPPs, with implications for applications such as spanning-tree-based preconditioners and graph sampling tasks.

Abstract

Given an $n\times r$ matrix $X$ of rank $r$, consider the problem of sampling $r$ integers $\mathtt{C}\subset \{1, \dots, n\}$ with probability proportional to the squared determinant of the rows of $X$ indexed by $\mathtt{C}$. The distribution of $\mathtt{C}$ is called a projection determinantal point process (DPP). The vanilla classical algorithm to sample a DPP works in two steps, an orthogonalization in $\mathcal{O}(nr^2)$ and a sampling step of the same cost. The bottleneck of recent quantum approaches to DPP sampling remains that preliminary orthogonalization step. For instance, (Kerenidis and Prakash, 2022) proposed an algorithm with the same $\mathcal{O}(nr^2)$ orthogonalization, followed by a $\mathcal{O}(nr)$ classical step to find the gates in a quantum circuit. The classical $\mathcal{O}(nr^2)$ orthogonalization thus still dominates the cost. Our first contribution is to reduce preprocessing to normalizing the columns of $X$, obtaining $\mathsf{X}$ in $\mathcal{O}(nr)$ classical operations. We show that a simple circuit inspired by the formalism of Kerenidis et al., 2022 samples a DPP of a type we had never encountered in applications, which is different from our target DPP. Plugging this circuit into a rejection sampling routine, we recover our target DPP after an expected $1/\det \mathsf{X}^\top\mathsf{X} = 1/a$ preparations of the quantum circuit. Using amplitude amplification, our second contribution is to boost the acceptance probability from $a$ to $1-a$ at the price of a circuit depth of $\mathcal{O}(r\log n/\sqrt{a})$ and $\mathcal{O}(\log n)$ extra qubits. Prepending a fast, sketching-based classical approximation of $a$, we obtain a pipeline to sample a projection DPP on a quantum computer, where the former $\mathcal{O}(nr^2)$ preprocessing bottleneck has been replaced by the $\mathcal{O}(nr)$ cost of normalizing the columns and the cost of our approximation of $a$.

Bypassing orthogonalization in the quantum DPP sampler

TL;DR

This work targets exact sampling of projection DPPs on a quantum computer by removing the traditional QR preprocessing bottleneck. It introduces column normalization to form and uses Clifford loaders to load the data, coupled with a rejection sampler whose acceptance is ; amplitude amplification and a sketch-based estimate of further reduce the number of rejections and overall cost. A key technical contribution is a novel DPP with a nonsymmetric (skew-symmetric) correlation kernel, exemplified by determinantal measures over dimer-rooted forests and connections to Pfaffians, which arises naturally in the graph-based instantiations and through the loading scheme. The numerical and implementation results in Qiskit demonstrate feasibility and offer a path toward practical quantum-speedups for sampling projection DPPs, with implications for applications such as spanning-tree-based preconditioners and graph sampling tasks.

Abstract

Given an matrix of rank , consider the problem of sampling integers with probability proportional to the squared determinant of the rows of indexed by . The distribution of is called a projection determinantal point process (DPP). The vanilla classical algorithm to sample a DPP works in two steps, an orthogonalization in and a sampling step of the same cost. The bottleneck of recent quantum approaches to DPP sampling remains that preliminary orthogonalization step. For instance, (Kerenidis and Prakash, 2022) proposed an algorithm with the same orthogonalization, followed by a classical step to find the gates in a quantum circuit. The classical orthogonalization thus still dominates the cost. Our first contribution is to reduce preprocessing to normalizing the columns of , obtaining in classical operations. We show that a simple circuit inspired by the formalism of Kerenidis et al., 2022 samples a DPP of a type we had never encountered in applications, which is different from our target DPP. Plugging this circuit into a rejection sampling routine, we recover our target DPP after an expected preparations of the quantum circuit. Using amplitude amplification, our second contribution is to boost the acceptance probability from to at the price of a circuit depth of and extra qubits. Prepending a fast, sketching-based classical approximation of , we obtain a pipeline to sample a projection DPP on a quantum computer, where the former preprocessing bottleneck has been replaced by the cost of normalizing the columns and the cost of our approximation of .

Paper Structure

This paper contains 26 sections, 13 theorems, 102 equations, 11 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Lighter preprocessing and rejection sampling algorithm.
  4. Reducing rejection rate with amplitude amplification.
  5. A DPP with non-symmetric correlation matrix.
  6. Implementation and numerics.
  7. Notations
  8. Sampling a projection DPP without orthogonalization
  9. CAR algebra and Clifford loaders
  10. State preparation with Clifford loaders
  11. Observing occupation numbers
  12. A rejection sampling algorithm
  13. Amplitude amplification to reduce the number of rejections
  14. Coherent Hamming measurements
  15. Amplitude amplification
  16. ...and 11 more sections

Key Result

Lemma 2.2

Let $Y$ be the point process defined in eq:def_Y. We have

Figures (11)

  • Figure 1: Effect of $\epsilon$-approximation of $a$ for amplitude amplification. On the left-hand side, we report the number of Grover iterations \ref{['eq:m_epsilon']} for boosting the acceptance probability as in \ref{['thm:guarantee_nb_Grover_epsilon']}. On the right-hand side, we display the lower bound \ref{['eq:lower_bound_epsilon']} on the acceptance probability of \ref{['a:rejection_sampling_Clifford_amplified_approx']}.
  • Figure 2: A simple square graph and the corresponding Clifford loader $\mathcal{C}(\mathsf{B}_{:1}) \mathcal{C}(\mathsf{B}_{:2})\mathcal{C}(\mathsf{B}_{:4})\ket{\emptyset}$; see \ref{['eq:def_psi']}. We here defined the normalized incidence matrix $\mathsf{B} = B D^{-1/2}$ and the notation $f_e = c_e + c_e^*$ for any edge $e$. The column $\varrho=3$ is removed from $\mathsf{B}$. The colored edges emanate from the node of the same color whereas their orientation is determined (by convention) by the matrix $B$.
  • Figure 3: Orientation associated to the matrix $\mathrm{skew}(L_{\widehat{\varrho}\widehat{\varrho}})$ for a simple graph. Edge orientations are determined by the ordering of the nodes. Dashed edges are missing after removing $\varrho$.
  • Figure 4: An augmented graph covered by a dimer configuration (thick edges). Here, the edges in $\mathtt{C} = \{(1,2),(2,3),(3,\varrho),(11,\varrho),(10,9)\}$ are depicted as teal rectangles whereas the perfect matching spanning the remaining nodes is colored in blue. The nodes of the original graph are displayed as circles.
  • Figure 5: A dimer-rooted forest with root $\varrho$ and whose edges -- colored in teal -- are $\mathtt{C} = \{(1,2),(2,3),(3,\varrho),(11,\varrho),(10,9)\}$. This dimer-rooted forest is associated with the augmented graph of \ref{['fig:augmented_graph']}.
  • ...and 6 more figures

Theorems & Definitions (33)

  • Remark 2.1: Complexity of a Clifford loader
  • Lemma 2.2: Law of conditioned process
  • proof
  • Remark 3.1: Rule of thumb for the quadratic speedup
  • Theorem 3.2: Effect of an $\epsilon$-approximation on number of Grover steps
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: Oracle for Grover's algorithm
  • proof
  • ...and 23 more