Gluing Theorem for Random Unitaries

• Gluing theorem for random unitaries defines a method to combine independent, Haar-sampled operations ensuring the composite mimics true Haar randomness even under inverse queries.
• It leverages path-recording purification and projector-based entanglement checks to achieve negligible trace distance from ideal randomness.
• The theorem underpins advances in quantum circuit complexity and cryptography by enabling efficient simulation and resource-stretched strong pseudorandom unitary constructions.

The gluing theorem for random unitaries articulates rigorous methods for composing random unitary operations—often independently sampled from the Haar measure—so that the resulting composite operation retains essential randomness and indistinguishability properties even in complex algorithmic or cryptographic settings. This concept is central to advances in quantum circuit complexity, quantum cryptography, information scrambling, and the efficient simulation of Haar-like randomness with low-depth or resource-efficient architectures.

1. Foundational Concepts and Theorem Statement

The gluing theorem asserts that multiple random unitaries, when properly composed (for example, via overlapping subsystems or controlled purification structures), yield a composite unitary whose output distribution is statistically close to Haar random, not merely for forward queries (standard oracle access) but also under inverse query access (adversary can query both $U$ and $U^\dagger$). Specifically, for unitaries $U_1, U_2, U_3$ acting on overlapping subsystems $A,B,C$, the glued unitary

$$G(U_1, U_2, U_3)_{ABC} = U_{3,BC}\, U_{2,AB}\, U_{1,AB}$$

constitutes a Haar random unitary on $ABC$ up to negligible trace distance for any polynomial-time query algorithm, including those permitted inverse queries. The security statement is

$$\operatorname{TD}\Bigl(\mathcal{A}\bigl[G(U_1,U_2,U_3)\bigr],\, \mathcal{A}[O]\Bigr) \le \text{negl}(n)$$

where $O$ denotes a Haar random unitary and $\mathcal{A}$ a $t$-query adversary (Ananth et al., 5 Oct 2025).

2. Purification, Path Recording, and Security under Inverse Queries

Central to the proof is the path-recording purification strategy. Each query to a random unitary results in the creation of an almost maximally entangled pair, with one half stored in an external purification database and the other used to produce the query response. The gluing process involves systematically recording the paths through which input states traverse different unitaries.

An alternate gluing method structures the purification into several disjoint or overlapping subspaces, adapting the database to accurately reflect entanglement and query history. The glued path-recording operator $V_{\text{glued}}$ (and its refined version $W_{\text{glued}}$) manages the query space in four distinct ways, depending on the observed entanglement structure. Crucially, this framework ensures that even when adversaries issue inverse queries (i.e., call $U^\dagger$), each branch of computation is correctly reflected in the purification space.

Security analysis employs projectors such as $\Pi_{IR,1}$ or $\Pi_{IT,1}$ to ascertain whether the purification subspaces properly capture the maximal entanglement required. The trace norm difference is bounded as

$$\Bigl\Vert \bigl(V_{\text{glued}} - V_{\text{ideal}}\bigr) \Pi_{\text{good}} \Bigr\Vert_{\mathrm{op}} \le O\Bigl(\frac{t^2}{2^{n/8}}\Bigr)$$

with the “good” subspace corresponding to properly structured purification (Ananth et al., 5 Oct 2025).

3. Randomness Stretching in Strong Pseudorandom Unitary Families

A principal application is the extension (or “stretching”) of strong pseudorandom unitaries (PRUs)—those indistinguishable from Haar random by algorithms with both forward and inverse access—with sublinear key expansion. Starting from a base strong PRU with key size $k$, repeated application of the gluing construction yields PRUs on $n$ qubits using only $O(n^{1/c})$ bits of randomness, for any constant $c$.

For instance, the composite construction

$$\text{StretchPRU}_{k_1|k_2|\dots|k_9} =(X_{k_3}\,\mathrm{PRU}_{k_2}\,X_{k_1})_{AB}\cdot(X_{k_6}\,\mathrm{PRU}_{k_5}\,X_{k_4})_{BC}\cdot(X_{k_9}\,\mathrm{PRU}_{k_8}\,X_{k_7})_{AB}$$

utilizes a cascade of base PRUs and random local transformations $X_{k_i}$, with seed lengths that increase only additively at each step (Ananth et al., 5 Oct 2025). Recursive gluing over $c$ levels leads to seed lengths of $O(n^{1/c})$ for $n$-qubit PRUs.

4. Technical Mechanisms: Projectors, Entanglement, and Monogamy Constraints

The gluing construction deploys projectors that check for local maximal entanglement. Monogamy-of-entanglement constraints ensure that entanglement spread across the glued units does not compromise the randomness properties when combined. Each query—whether forward or backward (inverse)—modifies the database in a way that is trace-preserving and maintains coupling between all relevant subspaces.

Progress measures track adversarial advantage; since entanglement and database structure are enforced after each query, the cumulative security error remains exponentially small in $n$, provided the number of queries $t$ grows no faster than a small polynomial.

5. Impact on Quantum Cryptography and Efficient Simulation

The gluing theorem’s assurance of indistinguishable composite Haar randomness under inverse query access allows cryptographic primitives such as random oracles, commitment schemes, and quantum-secure encryption to be instantiated with less randomness, lower-depth circuits, and robust resistance against reversal attacks.

Moreover, the result provides efficient simulation of Haar-like randomness: it is possible to “simulate” the effect of a large Haar random unitary by gluing several smaller ones—each with independent randomness and limited overlap—while matching indistinguishability metrics required in complexity theory and cryptographic analysis.

Comparison with prior constructions highlights the robustness of this approach. Previous techniques secured pseudorandom unitaries against only forward queries; the gluing theorem generically upgrades security to the strong (inverse query resistant) case, as evidenced by the negligible trace distance bounds for adversary outputs (Ananth et al., 5 Oct 2025).

6. Limitations and Interpretative Implications

The security proofs and randomness-stretching constructions rely on the existence of families of strong PRUs, which is contingent on underlying hardness assumptions (e.g., quantum-secure one-way functions). While the gluing technique affords key length reduction to $O(n^{1/c})$ for any constant $c$, practical implementation must ensure that the base PRU family itself does not leak information under composition, that the purification database scales efficiently, and that circuit depth remains operationally feasible.

A plausible implication is that cryptographic and complexity-theoretic models requiring Haar random features—such as indistinguishability obfuscation or quantum money—can now substitute true Haar sampling with efficiently glued PRU constructions, provided these structural and seed efficiency constraints are satisfied.

7. Summary Table: Gluing Theorem Features

Feature	Description	Security Guarantee
Number of Haar random unitaries	$O(1)$ (e.g., 3 for $ABC$ system)	Indistinguishability in trace norm
Allowed adversary access	Forward and inverse unitary queries	Robust (strong) PRU
Seed (key) stretching	$O(n^{1/c})$ random bits for $n$ qubits, any $c$	Negligible statistical distance from Haar
Purification method	Path-recording, monogamy tested by projectors	Structured database, no leakage
Circuit depth	Near-linear, preserves maximal entanglement per step	Efficient construction

These properties collectively formalize the rigorous composition of random unitaries with inverse query resistance and resource-efficient pseudorandomness, advancing both theoretical quantum information and practical quantum cryptographic applications.