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Halving the Cost of Controlled Time-Evolution

William A. Simon, Peter J. Love

TL;DR

The paper tackles the high resource burden of implementing controlled time evolution in fault-tolerant quantum simulation, where arbitrary rotations (non-Clifford gates) drive the dominant $T$-cost via magic-state consumption. It introduces a directionally-controlled compilation for symmetric Suzuki–Trotter decompositions that eliminates the rotation-count increase typically associated with control, reducing the controlled-time-evolution cost to the same level as uncontrolled evolution for second-order and higher decompositions. Specifically, it shows that second-order Trotterization requires only $2L$ arbitrary rotations under control (instead of $4L$ in naïve schemes), with higher orders scaling as $2L 5^{(p/2)-1}$, while also discussing CNOT-gate implications. The result significantly lowers the fault-tolerant resource requirements for quantum simulation and highlights the ongoing need to optimize magic-state generation and circuit compilation, since the practical gains depend on the relative costs of these components.

Abstract

Quantum simulation is a promising application for quantum computing. Quantum simulation algorithms may require the ability to control the time evolution unitary. Naive techniques to control a unitary can substantially increase the required computational resources. A standard approach to controlling Trotterized time evolution doubles the number of single-qubit arbitrary rotations. Here, we describe a compilation scheme that does not increase the number of arbitrary rotations for symmetric Trotterizations, which applies to second-order and higher Suzuki-Trotter decompositions. This halves the number of arbitrary rotations required to implement controlled, Trotterized time evolution compared to the standard approach. Arbitrary rotations contribute significantly to resource estimates in a fault-tolerant architecture due to the number of required magic states. Therefore, arbitrary rotations dominate the $T$-cost of fault-tolerant implementations of quantum simulation. This construction reduces the number of arbitrary rotations for controlled Trotter evolution to that of uncontrolled Trotter evolution, thereby reducing the cost of fault-tolerant quantum simulation.

Halving the Cost of Controlled Time-Evolution

TL;DR

The paper tackles the high resource burden of implementing controlled time evolution in fault-tolerant quantum simulation, where arbitrary rotations (non-Clifford gates) drive the dominant -cost via magic-state consumption. It introduces a directionally-controlled compilation for symmetric Suzuki–Trotter decompositions that eliminates the rotation-count increase typically associated with control, reducing the controlled-time-evolution cost to the same level as uncontrolled evolution for second-order and higher decompositions. Specifically, it shows that second-order Trotterization requires only arbitrary rotations under control (instead of in naïve schemes), with higher orders scaling as , while also discussing CNOT-gate implications. The result significantly lowers the fault-tolerant resource requirements for quantum simulation and highlights the ongoing need to optimize magic-state generation and circuit compilation, since the practical gains depend on the relative costs of these components.

Abstract

Quantum simulation is a promising application for quantum computing. Quantum simulation algorithms may require the ability to control the time evolution unitary. Naive techniques to control a unitary can substantially increase the required computational resources. A standard approach to controlling Trotterized time evolution doubles the number of single-qubit arbitrary rotations. Here, we describe a compilation scheme that does not increase the number of arbitrary rotations for symmetric Trotterizations, which applies to second-order and higher Suzuki-Trotter decompositions. This halves the number of arbitrary rotations required to implement controlled, Trotterized time evolution compared to the standard approach. Arbitrary rotations contribute significantly to resource estimates in a fault-tolerant architecture due to the number of required magic states. Therefore, arbitrary rotations dominate the -cost of fault-tolerant implementations of quantum simulation. This construction reduces the number of arbitrary rotations for controlled Trotter evolution to that of uncontrolled Trotter evolution, thereby reducing the cost of fault-tolerant quantum simulation.

Paper Structure

This paper contains 4 sections, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Trotterization
  3. Optimized Compilation
  4. Conclusion

Figures (3)

  • Figure 1: Pauli Operator Time Evolution.(a) The general structure of a circuit implementing the time evolution unitary generated by a $k$-local Pauli operator is shown. (b) The standard circuit for implementing the controlled time evolution unitary generated by a $k$-local Pauli operator is shown. This circuit requires two arbitrary rotations.
  • Figure 2: Directionally-Controlled Time Evolution.(a) The circuit for directionally controlling the time evolution operator generated by a multi-qubit Pauli operator only requires two additional CNOT gates compared to the uncontrolled operation. (b) For symmetric Trotterizations, directional control can be implemented by directionally controlling the time evolution of each term.
  • Figure 3: Controlled Time Evolution.(a) The controlled time evolution of a second-order Suzuki-Trotter decomposition can be achieved using only $2L$ rotations, whereas the naive construction shown in Figure \ref{['fig:pauli-evolution']} requires $4L$ rotations. (b) Controlled time evolution for higher-order Suzuki-Trotter decompositions can be implemented with fewer CNOT gates by using uncontrolled and directionally controlled components.