Reversible Lifetime Semantics for Quantum Programs

Abstract

Reversible computation requires that intermediate data be explicitly undone rather than discarded. In quantum programming, this principle appears as uncomputation, usually treated as a technical cleanup mechanism. We instead present uncomputation as a semantic foundation. In the Qutes language, we introduce a formal model of \emph{Scope-Bounded Liveness-Guided Uncomputation}, where lexical scope bounds variable lifetime and static liveness and entanglement analysis determine the earliest safe reclamation point. We define semantic lifetime and a Restoration Invariant ensuring that temporary quantum information disappears once it becomes semantically irrelevant. We prove compositional correctness under nested scopes and show that early reclamation can reduce circuit depth by avoiding critical-path overhead and can bound peak live qubits through disciplined ancilla reuse. Finally, we show that parameter passing semantics emerges from the same lifetime discipline, with pass-by-value and pass-by-reference corresponding to different lifetime boundaries, and we characterize the constraints (irreversibility, persistent entanglement, and aliasing) under which automatic uncomputation must be restricted.

Figures (5)

• Figure 1: Circuit-level realization of the running example. The forward computation (left) implements the unitary blocks $U_f$, $U_g$, $U_h$, and $U_k$, while the rightmost segment illustrates the classical compute--then--uncompute strategy in which adjoint blocks are appended at the end of the circuit.
• Figure 2: (on the left) Data Dependence Graph for the example of Fig. \ref{['fig:running-example']}. Solid arrows denote functional data dependencies, while dashed bidirectional edges represent entanglement-induced dependencies.(on the right) Entanglement Graph for the example: dashed edges connect variables that may become entangled through controlled operations.
• Figure 3: Semantic lifetimes under early uncomputation of the example in Fig. \ref{['fig:running-example']}. While entanglement-based liveness conservatively extends $t_1$ and $t_2$ to the end of the program, the subcircuit preceding $p_4$ is output-isolable; hence $t_1$ and $t_2$ can be safely reclaimed after $p_4$, whereas $t_3$ and $t_4$ remain live until $p_9$ due to their interaction with $y_2$.
• Figure 4: Circuit with early uncomputation: the adjoint blocks associated with $(t_1,t_2)$ are inserted immediately after their last semantic use, allowing overlap with the subsequent computation and reducing the overall circuit depth compared to global compute--then--uncompute.
• Figure 5: Width-reduced circuit obtained via early reclamation: the temporaries $t_1$ and $t_2$ are restored immediately after their last semantic use, preventing overlap with $t_3$ and $t_4$ and reducing the peak number of simultaneously live qubits.

Theorems & Definitions (30)

• definition 1: Data Dependence Graph
• definition 2: Entanglement Graph
• definition 3: Liveness
• definition 4: Semantic Lifetime
• definition 5: Output-Isolable Subcircuit
• definition 6: Safe Reclamation Point
• definition 7: Critical Path
• lemma 1: Depth Bound under Early Reclamation
• proof
• lemma 2: Zero-Overhead Condition