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Limitations of Noisy Reversible Computation

D. Aharonov, M. Ben-Or, R. Impagliazzo, N. Nisan

TL;DR

The paper investigates the computational power when both noise and reversibility are present, showing that the joint model is weaker than unrestricted computation. It establishes a classical upper bound where any circuit can be simulated by a noisy reversible circuit with size $O(s 2^{O(d)})$ and depth $O(d)$, and a matching lower bound showing that non-trivial circuits require size $s$ exponential in depth, placing noisy reversible computation at the level of $NC^1$. The quantum extension yields analogous upper bounds and a non-tight lower bound, indicating that polynomial-size noisy quantum circuits are not stronger than $QNC^1$ and that quasi-polynomial circuits can simulate $QNC^1$, with open questions about tightness and constant-depth noise resistance. Together, these results highlight how modest noise coupled with reversibility dramatically constrains computational power and underscore the sensitivity of model definitions in determining complexity power.

Abstract

Noisy computation and reversible computation have been studied separately, and it is known that they are as powerful as unrestricted computation. We study the case where both noise and reversibility are combined and show that the combined model is weaker than unrestricted computation. In our noisy reversible circuits, each wire is flipped with probability p each time step, and all the inputs to the circuit are present in time 0. We prove that any noisy reversible circuit must have size exponential in its depth in order to compute a function with high probability. This is tight as we show that any circuit can be converted into a noise-resistant reversible one with a blow up in size which is exponential in the depth. This establishes that noisy reversible computation has the power of the complexity class NC^1. We extend this to quantum circuits(QC). We prove that any noisy QC which is not worthless, and for which all inputs are present at time 0, must have size exponential in its depth. (This high-lights the fact that fault tolerant QC must use a constant supply of inputs all the time.) For the lower bound, we show that quasi-polynomial noisy QC are at least powerful as logarithmic depth QC, (or QNC^1). Making these bounds tight is left open in the quantum case.

Limitations of Noisy Reversible Computation

TL;DR

The paper investigates the computational power when both noise and reversibility are present, showing that the joint model is weaker than unrestricted computation. It establishes a classical upper bound where any circuit can be simulated by a noisy reversible circuit with size and depth , and a matching lower bound showing that non-trivial circuits require size exponential in depth, placing noisy reversible computation at the level of . The quantum extension yields analogous upper bounds and a non-tight lower bound, indicating that polynomial-size noisy quantum circuits are not stronger than and that quasi-polynomial circuits can simulate , with open questions about tightness and constant-depth noise resistance. Together, these results highlight how modest noise coupled with reversibility dramatically constrains computational power and underscore the sensitivity of model definitions in determining complexity power.

Abstract

Noisy computation and reversible computation have been studied separately, and it is known that they are as powerful as unrestricted computation. We study the case where both noise and reversibility are combined and show that the combined model is weaker than unrestricted computation. In our noisy reversible circuits, each wire is flipped with probability p each time step, and all the inputs to the circuit are present in time 0. We prove that any noisy reversible circuit must have size exponential in its depth in order to compute a function with high probability. This is tight as we show that any circuit can be converted into a noise-resistant reversible one with a blow up in size which is exponential in the depth. This establishes that noisy reversible computation has the power of the complexity class NC^1. We extend this to quantum circuits(QC). We prove that any noisy QC which is not worthless, and for which all inputs are present at time 0, must have size exponential in its depth. (This high-lights the fact that fault tolerant QC must use a constant supply of inputs all the time.) For the lower bound, we show that quasi-polynomial noisy QC are at least powerful as logarithmic depth QC, (or QNC^1). Making these bounds tight is left open in the quantum case.

Paper Structure

This paper contains 13 sections, 17 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Noisy reversible circuits - The Upper Bound
  3. Noisy Reversible Circuits - The Lower Bound
  4. Quantum Computation
  5. Pure states
  6. Mixed states
  7. Quantum circuits with mixed states
  8. Noisy Quantum Circuits
  9. Quantum Entropy
  10. The Quantum upper Bound
  11. The Quantum Lower Bound
  12. Open Question
  13. Appendix-Quantum Entropy lemmas

Key Result

Proposition 1

If a boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ can be computed by a circuit of size $s$ and depth $d$, then it can be implemented by reversible circuits of size $O(s)$ and depth $O(d)$.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Theorem 1
  • Definition 3
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 4
  • Corollary 2
  • ...and 13 more