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Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers

Shantanav Chakraborty

TL;DR

Three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications, are developed, each requiring only a single ancilla qubit, and no multi-qubit controlled operations.

Abstract

We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.

Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers

TL;DR

Three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications, are developed, each requiring only a single ancilla qubit, and no multi-qubit controlled operations.

Abstract

We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.
Paper Structure (36 sections, 32 theorems, 301 equations, 3 figures, 7 tables, 6 algorithms)

This paper contains 36 sections, 32 theorems, 301 equations, 3 figures, 7 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Summary of our results
  3. Single-Ancilla LCU: Estimating expectation values of observables
  4. Analog LCU: Coupling discrete systems with continuous variable systems
  5. Ancilla - free LCU: Randomized sampling of unitaries
  6. Prior work
  7. Preliminaries
  8. Notation
  9. Linear Combination of Unitaries
  10. Block encoding and Quantum Singular Value Transformation
  11. Three different approaches for implementing LCU on intermediate-term quantum computers
  12. Robustness of expectation values of observables
  13. Single-Ancilla LCU: Estimating expectation values of observables
  14. Comparison with the standard LCU procedure
  15. Analog LCU: coupling a discrete primary system to a continuous-variable ancilla
  16. ...and 21 more sections

Key Result

Theorem 1

Let $O$ be some observable and $\rho_0$ be some initial state, prepared in cost $\tau_{\rho_0}$. Suppose there exists a Hermitian matrix $H\in\mathbb{C}^{N\times N}$, and a function $f:[-1,1]\mapsto\mathbb{R}$ such that $f(H)\approx \sum_{j} c_j U_j$, where $\norm{c}_1=\sum_{j}|c_j|$, and each $U_j$ with a constant probability, using only one ancilla qubit and repetitions of the quantum circuit i

Figures (3)

  • Figure 1: Summary of the main results -- The three approaches to LCU and their applications
  • Figure 2: Quantum circuit corresponding for the Single-Ancilla LCU procedure. For $f(H)\approx \sum_{j}c_j U_j$, repeated runs of this short-depth quantum circuit can estimate $\mathrm{Tr}[O~f(H)\rho_0 f(H)^{\dag}]/\mathrm{Tr}[f(H)\rho_0 f(H)^{\dag}]$, to arbitrary accuracy. For this, $V_1$ and $V_2$ are sampled at random according to $\mathcal{D}\sim \{c_j/\norm{c}_1, U_j\}$. Each run of the circuit outputs a random variable corresponding to the outcome of the measurement of the observable $X\otimes O$. Overall we need to repeat this circuit $T$ times, such that the sample mean of the $T$ observations to converge to the desired estimate.
  • Figure 3: Relationship between discrete and continuous-time quantum walks, and their classical counterparts: Given the block encoding of the discriminant matrix $D$ of any reversible Markov chain $P$, we can generate a continuous-time quantum walk on both the vertices and edges of $P$, following Ref. childs2010relationship. Conversely, given access to the continuous-time quantum time-evolution operator $U=e^{-iH/2}$, we can implement a discrete-time quantum walk on the edges of $H$. For this, we make use of the framework of Quantum Singular Value Transformation (QSVT). Apers and Sarlette demonstrated that discrete-time random walks can be fast forwarded by discrete-time quantum walks apers2018FF which is the cornerstone of the recently developed optimal quantum spatial search algorithm ambainis2019quadratic. In this work, we show that for quantum spatial search, the fast-forwarding can be done without needing any ancilla qubits (other than the quantum walk space) through Ancilla-free LCU. Additionally, we also demonstrate that discrete-time quantum walks can fast forward continuous-time random walks. This fact, can also be leveraged to develop new optimal quantum spatial search algorithms, which do not require any ancilla qubits (other than the quantum walk space). Finally continuous-time random walks can also be fast-forwarded by continuous-time quantum walks which is central to the optimal quantum spatial search algorithm of apers2022quadratic. Thus overall, through this work we connect all different frameworks random and quantum walks.

Theorems & Definitions (55)

  • Theorem 1: Informal
  • Theorem 2: Randomized unitary sampling
  • Definition 3: Block Encoding chakraborty2019power
  • Theorem 4: Quantum Singular Value Transformation gilyen2018quantum
  • Lemma 5: Robustness of Quantum Singular Value Transformation, gilyen2019quantum
  • Lemma 6: Tracial version of Hölder's inequality ruskai1972inequalities
  • Theorem 7
  • proof
  • Theorem 8: Estimating expectation values of observables
  • proof
  • ...and 45 more