Quantum Algorithm for Reversing Unknown Unitary Evolutions

TL;DR

The Quantum Unitary Reversal Algorithm (QURA), a deterministic and exact approach to universally reverse arbitrary unknown unitary transformations using $\mathcal{O}(d^2)$ calls of the unitary, where $d$ is the system dimension.

Abstract

Reversing an unknown quantum evolution is of central importance to quantum information processing and fundamental physics, yet it remains a formidable challenge as conventional methods necessitate an infinite number of queries to fully characterize the quantum process. Here we introduce the Quantum Unitary Reversal Algorithm (QURA), a deterministic and exact approach to universally reverse arbitrary unknown unitary transformations using $\mathcal{O}(d^2)$ calls of the unitary, where $d$ is the system dimension. Our quantum algorithm resolves a fundamental problem of time-reversal simulations for closed quantum systems by confirming the feasibility of reversing any unitary evolution without knowing the exact process. The algorithm also provides the construction of a key oracle for unitary inversion in many quantum algorithm frameworks, such as quantum singular value transformation. It notably reveals a sharp boundary between the quantum and classical computing realms and unveils a quadratic quantum advantage in computational complexity for this foundational task.

Figures (8)

• Figure 1: Schematic depiction of Quantum Unitary Reversal Algorithm. (a). A quantum computer could simulate the inverse of an unknown unitary evolution by querying it for finite times. (b). QURA, akin to the idea of amplitude amplification, is structured in three stages (1), (2), and (3). The input quantum state $|\varphi\rangle_d$ is initialized into the superposition of the target $U^{\dag} |\varphi\rangle_d$ and the other unwanted components with averaged amplitudes. A duality-based amplitude amplifier ${\cal A}_{\alpha=1}$ is used iteratively to enhance the amplitude of the target state with a constant angle $\Delta_d$. The iteration ends when the angle approaches its maximum before surpassing $\pi/2$. Consequently, a tunable amplifier, denoted as ${\cal A}_{\alpha=\delta}$, completes the angular amplification up to $\pi/2$, giving rise to the inverse of unitary with all ancillary qudits returning to zero states.
• Figure 2: An abstract depiction of one rotation of angle $\Delta_d$, between two vector subspaces in ${{\mathbb C}}^{2d^3}$ determinated by vectors $|\Psi_0\rangle$, $|\Lambda_0\rangle$ and unitary $E (U)$. Here $\widetilde{ D }(U)$ is a unitary analog to $E (U)^\dag$, constructed by $D (U)$.
• Figure S1: Depiction of the rotations between the two dual subspaces, where the integrated rotation $E\Tilde{D}$ achieves a $\Delta$ angle amplifying to the target $\Psi_0$.
• Figure S2: Depiction of Initialization and first round of amplitude amplify. The state starts from $\Lambda_0$ and after $E \cdot \widetilde{ D } \cdot E$, we reach the state $E \widetilde{ D } E \Lambda_0$ with $\sin(2\Delta)$ amplitude component of the target state $\Psi_0$.
• Figure S3: Quantum circuit representation of the amplifier ${\cal A}_{\alpha}$ with respect to the definition in Eq \ref{['eqn:mod amp']}.

Theorems & Definitions (14)

• Theorem 1
• Theorem \ref{thm:unitary inv cir exist}
• Lemma S2
• Lemma S3
• Lemma S4: Dual relations
• Corollary S5: Dual relations, simplified
• Lemma S6
• Lemma S7
• Lemma S8
• Lemma S9