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Revision #1 to TR20-087 | 23rd June 2020 18:12

#### Quantum Logspace Algorithm for Powering Matrices with Bounded Norm

Revision #1
Authors: Uma Girish, Ran Raz, Wei Zhan
Accepted on: 23rd June 2020 18:12
Keywords:

Abstract:

We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary $n\times n$ contraction matrix $A$, and a parameter $T \leq poly(n)$ and outputs the entries of $A^T$, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result:

First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space $O(S + \log T)$ that takes as an input the description of a quantum algorithm with quantum space $S$ and time $T$, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements.

Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace.

Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.

### Paper:

TR20-087 | 8th June 2020 21:49

#### Quantum Logspace Algorithm for Powering Matrices with Bounded Norm

TR20-087
Authors: Uma Girish, Ran Raz, Wei Zhan
Publication: 8th June 2020 21:51
Keywords:

Abstract:

We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary $n\times n$ contraction matrix $A$, and a parameter $T \leq poly(n)$ and outputs the entries of $A^T$, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result:

First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space $O(S + \log T)$ that takes as an input the description of a quantum algorithm with quantum space $S$ and time $T$, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements.

Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace.

Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.

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