Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection

TL;DR

This work introduces rewinding operators that invert quantum measurements and defines three complexity classes—$RwBQP$, $CBQP$, and $AdPostBQP$—to study their computational power. It proves a central equivalence and containment: $BPP^{PP} \subseteq RwBQP = CBQP = AdPostBQP \subseteq PSPACE$, with $AdPostBQP$ also contained in PSPACE and a mitigation protocol equating rewinding with postselection under polynomial overhead. The paper further explores restricted models, showing that rewindable Clifford circuits remain classically simulatable, while rewindable IQP circuits achieve PP-level power, and demonstrates that a single rewinding operator suffices for certain cryptographic-hard tasks under SIVP assumptions. Overall, rewinding provides a unified lens to compare cloning, adaptive postselection, and classical simulability, with implications for quantum cryptography, complexity borders, and the practicality of postselection-like techniques.

Abstract

We define rewinding operators that invert quantum measurements. Then, we define complexity classes ${\sf RwBQP}$, ${\sf CBQP}$, and ${\sf AdPostBQP}$ as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that ${\sf BPP}^{\sf PP}\subseteq{\sf RwBQP}={\sf CBQP}={\sf AdPostBQP}\subseteq{\sf PSPACE}$. As a byproduct of this result, we show that any problem in ${\sf PostBQP}$ can be solved with only postselections of events that occur with probabilities polynomially close to one. Under the strongly believed assumption that ${\sf BQP}\nsupseteq{\sf SZK}$, or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. Finally, we show that rewindable Clifford circuits remain classically simulatable, but rewindable instantaneous quantum polynomial time circuits can solve any problem in ${\sf PP}$.

Figures (2)

• Figure 1: Concrete example of ${\sf RwBQP}$ computation. $\mathcal{D}$, $U$, $R$, and $|\psi\rangle$ are a classical description of $U|00\rangle$, a two-qubit unitary operator, the rewinding operator, and the output state, respectively. More precisely, when $U=\prod_iu_i$ for elementary quantum gates $u_i$ in a universal gate set, $\mathcal{D}$ is a bit string representing $\prod_iu_i$. Note that since $|\psi\rangle$ is prepared by using only the unitary operator $U$, its classical description $\mathcal{D}$ does not include projectors and can be generated from only $\tilde{\mathcal{D}}$. Meter symbols represent the Pauli-$Z$ measurements, and $s_i\in\{0,1\}$ is the $i$th measurement outcome for $1\le i\le 3$. We represent $|s_i\rangle$ as a classical bit $s_i$ to emphasize that it can be copied. When the first measurement outcome $s_1$ is $1$, the first rewinding operator $R$ is applied. On the other hand, when $s_1=0$, we do not apply $R$, because the target state is obtained. Since the second and third measurements and the second rewinding operator are applied only when $s_1=1$, they are also conditioned on $s_1$. In a similar way, since it is not necessary to apply the second rewinding operator if $s_2=0$, the second rewinding operator and the third measurement are also conditioned on $s_2$. Finally, $|\psi\rangle$ becomes the target state when $s_1=0$, $(s_1,s_2)=(1,0)$, or $(s_1,s_2,s_3)=(1,1,0)$.
• Figure 4: Quantum circuit diagram of our mitigation protocol. $R$ represents the rewinding operator, and $|\phi\rangle=\alpha'|0\rangle_2|\psi_0\rangle+\beta'|1\rangle_2|\psi_1\rangle$. The red wire represents the second qubit of $U_x|0^n\rangle$ that will be postselected.

Theorems & Definitions (30)

• Definition 1: ${\sf PostBQP}$ A05
• Definition 2: ${\rm SIVP}_\gamma$
• Theorem 3: adapted from CCKW18
• Definition 4: Rewinding and Cloning Operators
• Definition 5: ${\sf RwBQP}$ and ${\sf CBQP}$
• Lemma 6
• Definition 7: ${\sf AdPostBQP}$
• Remark 8
• Corollary 9
• Corollary 10