Quantum machine learning with subspace states

Kerenidis, Iordanis; Prakash, Anupam

Quantum machine learning with subspace states

Jan 31, 2022

32 pages

e-Print:

2202.00054 [quant-ph]

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Abstract: (arXiv)

We introduce a new approach for quantum linear algebra based on quantum subspace states and present three new quantum machine learning algorithms. The first is a quantum determinant sampling algorithm that samples from the distribution

\Pr[S]= det(X_{S}X_{S}^{T})

for

|S|=d

using

O(nd)

gates and with circuit depth

O(d\log n)

. The state of art classical algorithm for the task requires

O(d^{3})

operations \cite{derezinski2019minimax}. The second is a quantum singular value estimation algorithm for compound matrices

\mathcal{A}^{k}

, the speedup for this algorithm is potentially exponential. It decomposes a

\binom{n}{k}

dimensional vector of order-

k

correlations into a linear combination of subspace states corresponding to

k

-tuples of singular vectors of

A

. The third algorithm reduces exponentially the depth of circuits used in quantum topological data analysis from

O(n)

to

O(\log n)

. Our basic tool are quantum subspace states, defined as

|Col(X)\rangle = \sum_{S\subset [n], |S|=d} det(X_{S}) |S\rangle

for matrices

X \in \mathbb{R}^{n \times d}

such that

X^{T} X = I_{d}

, that encode

d

-dimensional subspaces of

\mathbb{R}^{n}

. We develop two efficient state preparation techniques, the first using Givens circuits uses the representation of a subspace as a sequence of Givens rotations, while the second uses efficient implementations of unitaries

\Gamma(x) = \sum_{i} x_{i} Z^{\otimes (i-1)} \otimes X \otimes I^{n-i}

with

O(\log n)

depth circuits that we term Clifford loaders.

determinant
quantum algebra
unitarity
topological
rotation
Clifford
computer: quantum
data analysis method
numerical methods

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