Relative-Error Approximate Unitary Designs

• Relative-error approximate unitary designs are ensembles that emulate the Haar measure with multiplicative error bounds on low-degree moments, ensuring robust quantum behavior.
• Recent constructions using recursive crosstwirl, magic-augmented Clifford circuits, and blocked LRFC achieve efficient circuit depth and resource optimization.
• These designs underpin advancements in quantum cryptography, channel capacity theory, and simulation by providing operationally indistinguishable and error-resilient quantum protocols.

Relative-error approximate unitary designs are ensembles of unitary operators that provide strong, multiplicative-error approximations to the uniform (Haar) measure in terms of their action on low-degree moments. Such designs are central in quantum information science, as they enable efficient emulation of Haar-randomness with quantifiable, operationally meaningful guarantees that persist even under adaptive or arbitrary quantum experiments. Recent research has elucidated their construction, quantitative properties, circuit complexity, and applications, as well as the interplay between error notions, operational distinguishability, and structural constraints.

1. Definitions and Hierarchies of Error for Unitary Designs

A unitary $k$-design is an ensemble of unitaries $\mathcal{E}$ (possibly with probability distribution) such that, for any balancing degree-$k$ polynomial observable $O$ on the Hilbert space, the $k$-moment channel induced by the ensemble matches the Haar average on $U(d)$. Formally, letting

$$\Phi^{(k)}_{\mathcal{E}}(O) = \mathbb{E}_{U \sim \mathcal{E}} [ U^{\otimes k} O (U^\dagger)^{\otimes k} ],$$

the ensemble is an exact $k$-design if $$\Phi^{(k)}_{\mathcal{E}} = \Phi^{(k)}_{\mathrm{Haar}}$$.

There exists a hierarchy of approximate design definitions, crucially distinguished by the notion of error:

• Additive Error:

$$\|\Phi^{(k)}_{\mathcal{E}} - \Phi^{(k)}_{\mathrm{Haar}}\|_\diamond \leq \epsilon,$$

where $\|\cdot\|_\diamond$ is the diamond norm.

• Measurable Error: For any quantum protocol involving up to $k$ (possibly adaptive) queries,

$$\sup_{W} \left\| \mathbb{E}_{U \sim \mathcal{E}}[\rho_W^U] - \mathbb{E}_{U \sim \mathrm{Haar}}[\rho_W^U] \right\|_1 \leq \epsilon.$$

• Relative (Multiplicative) Error: (Strongest; also called multiplicative or multiplicative-approximate $k$-design)

$$(1-\epsilon) \Phi^{(k)}_{\mathrm{Haar}} \preceq \Phi^{(k)}_{\mathcal{E}} \preceq (1+\epsilon) \Phi^{(k)}_{\mathrm{Haar}}$$

where $\preceq$ denotes complete positivity ordering. This definition guarantees that the $k$-moment channel of $\mathcal{E}$ approximates the Haar average up to a multiplicative factor $1\pm\epsilon$ for all positive semidefinite inputs.

Relative error is strictly stronger than additive or measurable error: it implies operational indistinguishability even for postselection-based and adaptive protocols, and ensures all output eigenvalues and statistics are within relative error $\epsilon$. This is essential, for example, in derandomized quantum cryptography and derandomized proofs of superadditivity in channel capacities.

2. Constructions and Circuit Complexity

Recent research has established both Shannon-theoretic existence and explicit circuit constructions of relative-error approximate $k$-designs with resource requirements approaching various optimalities.

2.1. General Structural Methods

• Block-based Recursive (Crosstwirl) Schemes: By recursively applying local $k$-designs and periodically “crosstwirling” over small overlapping subsystems (interleaved layers of random unitaries over overlapping blocks), shallow circuits with patch size and circuit depth independent, or only logarithmically dependent, on global system size $m$ can efficiently realize relative-error $k$-designs. The key protocol steps are:
  1. Apply exact or approximate $k$-design unitaries independently to $O(\ell)$-qudit blocks.
  2. Swap a subset of $\ell$ qudits between neighboring blocks or perform crosstwirls.
  3. Iterate with logarithmically many overlapping layers (each acting on different partitions).

Depth Scaling: For $m$-qudit systems, circuit depth

$$D = O\left( (k\log k + \log m + \log(1/\epsilon))\, k \,\mathrm{polylog}(k) \right)$$

suffices for multiplicative error $\epsilon$ ($k$ fixed, $m$ large) (LaRacuente et al., 10 Jul 2024).

• Magic-augmented Clifford Circuits: Clifford circuits sandwiched between constant-depth layers (“magic gates”) composed of random $k$-design unitaries on $O(k\log k)$-qubit disjoint clusters can realize relative-error $k$-designs with circuit depth $O(\log(N/\epsilon)) + 2^{O(k\log k)}$ in 1D and $O(\log\log(N/\epsilon)) + 2^{O(k\log k)}$ in all-to-all (Zhang et al., 3 Jul 2025). For additive error, the number of magic gates can be made independent of $N$.
• Blocked LRFC (Luby-Rackoff-Function-Clifford) Circuits: Alternating blocks of highly explicit local random Clifford and hash-based mixing (using $k$-wise independent random functions to shuffle and relabel patchwise single- and two-qubit unitaries) lead to designs with measurable or relative error in depth

$O(\log k\, \log\log(nk/\epsilon))$

(for measurable error) or $O(k \log k\, \log\log(nk/\epsilon))$ (for multiplicative error), with near-optimal ancilla and randomness count (Cui et al., 8 Jul 2025).

2.2. Scaling and Resource Requirements

Construction	Error Notion	Circuit Depth	Patch/Swap Size	Ancilla	Random Bits
Recursive/crosstwirl (area law)	Relative	$$O([k\log k+\log m + \log(1/\epsilon)]\, k\,\mathrm{polylog}k)$$	$O(k\log k\!+\!\log(1/\epsilon))$	$O(m)$	$O(mk)$
Magic-augmented Clifford (1D)	Relative	$O(\log N\!+\!2^{O(k\log k)})$	$O(k\log k)$	$O(N)$	$O(N)$
Blocked LRFC	Measurable	$O(\log k\, \log \log(nk/\epsilon))$	$\log(nk/\epsilon)$	$O(nk)$	$O(nk)$
Blocked LRFC (amplified)	Relative	$O(k \log k\, \log\log(nk/\epsilon))$	$\log(nk/\epsilon)$	$O(nk)$	$O(nk)$
Prior (random local circuits, BHH)	Relative	$O(nk)$	$O(1)$	none	$O(nk)$

• Entanglement properties: These shallow constructions can guarantee that, for any region $S$, the bipartite communication (and thus the entanglement entropy) across its boundary $\partial S$ obeys an area law with subpolynomial corrections, in contrast to volume-law entanglement produced by Haar-random unitaries of full circuit depth (LaRacuente et al., 10 Jul 2024).
• Seed Length: Strongly explicit constructions with seed complexity $O(nk+\log(1/\epsilon))$ exist, matching random sampling up to polynomials (O'Donnell et al., 2023).

3. Analytical Methods: Error Conversion and Measurement-induced Distinguishability

3.1. Error Conversion via Von Neumann Index

Traditionally, conversion from $2$-norm (tensor product expander) bounds to relative error (complete positivity ordering) introduces a dimension-dependent penalty. Recent work leverages the theory of von Neumann algebras and subfactor indices to achieve dimension-free operator norm bounds for specific subalgebras associated with the $k$-fold twirl (e.g., Schur-Weyl fixed-point algebras underlying symmetric $k$-fold moments):

$$\| \Phi - E_N \|_{\mathrm{cb}} \leq \epsilon \implies (1-\epsilon)E_N \preceq \Phi \preceq (1+\epsilon)E_N$$

where $E_N$ is the projector onto the fixed-point algebra, and the subfactor index appears as a bounded multiplicative factor in the operator norm conversion, typically scaling as $k!^r$ for $r$ partitions (LaRacuente et al., 10 Jul 2024).

3.2. Alternating Projection Analysis

Protocols such as crosstwirl or twirl-swap-twirl are analyzed via the method of alternating projections onto the fixed-point subspaces of the local and global twirl channels. The speed of convergence (hence required patch size $\ell$) is controlled by the principal angles between these subspaces, precisely computed using Schur-Weyl duality and representation theory.

3.3. Measurement-based and Experimental Distinguishability

Constructed ensembles can be shown to have small measurable error: $$\sup_{W_{1\cdots k+1}}\left\| \mathbb{E}_{U\sim \mathcal{E}}[\rho_W^U] - \mathbb{E}_{U \sim \mathrm{Haar}}[\rho_W^U] \right\|_1 \leq \epsilon$$ where the supremum runs over all $k$-interleaved arbitrary (even adaptive) protocols and measurements; for applications such as randomized benchmarking, cryptography, and decoupling, this offers operationally robust guarantees beyond the parallel query/additive error regime (Cui et al., 8 Jul 2025).

4. Physical Implications and Applications

• Quantum supremacy and sampling hardness: Shallow, local circuits with unitary $k$-design properties can anti-concentrate and yield probability distributions for which approximate classical sampling is hard, matching the practical regime of superconducting quantum circuits (Harrow et al., 2018).
• Quantum channel and capacity theory: Relative-error $k$-designs are sufficient to demonstrate superadditivity of classical and minimum output Rényi capacities, previously provable only for Haar-random constructions (Nema et al., 2019).
• Quantum cryptography: Non-malleable encryption and derandomized information theoretic security protocols are achievable by (even existentially sampled) relative-error $k$-designs, with exponentially improved key and sample size versus full Haar measure (Lancien et al., 2019).
• Quantum thermodynamics and scrambling: High-probability decoupling, thermalization, and information mirroring (“Hayden-Preskill black hole”) phenomena are realizable with efficiently generated relative-error approximate designs, not requiring full Haar randomness (Nema et al., 2020).
• Quantum simulation and condensed matter: The area-law entanglement in these designs underpins the difference between entropy and randomness in shallow versus deep quantum circuits. This bridges the study of quantum pseudorandomness with resource constraints relevant in quantum many-body physics (LaRacuente et al., 10 Jul 2024).

5. Open Problems and Limitations

• Constant-size seeds and local gate sets: While constructions with growing local seed sets yield efficient relative-error designs, it remains an open problem to realize such designs from constant-size, physically natural gate sets for large $k$ and $n$ (Mezher et al., 2019, Lancien et al., 2019).
• No-go results for shallow circuits: Certain architectures (matrix product states, strictly finite-depth Clifford circuits without global magic augmentation, etc.) cannot achieve relative-error $k$-designs for $k\geq4$ unless circuit depth and magic content scale at least logarithmically with system size [(Zhang et al., 3 Jul 2025), Table 2].
• Amplification for relative error: For some constructions initially yielding only measurable (not multiplicative) error, an additional design amplification step (serial composition, depth $O(k)$) is required to reach the relative-error regime, though this remains polylogarithmic in $\epsilon$ and $n$ (Cui et al., 8 Jul 2025).
• Resource balancing: There is an inherent tradeoff between circuit depth, random seed length, ancilla usage, and the patch size required for localized randomization; each can be tuned within the analytic frameworks provided to match experimental or architectural constraints.

6. Summary Table: Circuit Complexity for Relative-Error Designs

Circuit Model	Relative Error?	Depth Scaling (for fixed $k$, $\epsilon$)	Ancilla Scaling	Comments
Recursive Crosstwirl	Yes	$$O\big((k\log k+\log m+\log(1/\epsilon))\,k\,\text{polylog}k\big)$$	$O(m)$	Area law, local 2-qudit gates (LaRacuente et al., 10 Jul 2024)
Magic-augmented Clifford	Yes	$O(\log N)+2^{O(k\log k)}$	$O(N)$	Strictly local magic for state designs (Zhang et al., 3 Jul 2025)
Blocked LRFC (amplified)	Yes	$O(k \log k\, \log \log (nk/\epsilon))$	$O(nk)$	Near-optimal ancilla/randomness, explicit (Cui et al., 8 Jul 2025)
Haar random circuits (BHH)	Yes	$O(nk)$	None	Lower bound matching for $1$D (Hunter-Jones, 2019)

7. Outlook and Research Directions

Advances in the construction of relative-error unitary designs have clarified the necessary circuit depth, locality, and randomness constraints for achieving operationally meaningful pseudorandomness. Key open directions include further optimizing patch/seed sizes, proving optimality for fixed and low-depth architectures, transferring these methods to higher local dimension and non-qubit systems, and exploring tradeoffs between area-law entanglement and randomization for complex quantum algorithms. The analytical frameworks—combining state/circuit decompositions, algebraic duality, and subfactor index theory—now enable tight, physically relevant analysis of quantum pseudorandomness in regimes encompassing both information-theoretic security and practical quantum simulation (LaRacuente et al., 10 Jul 2024, Cui et al., 8 Jul 2025, Zhang et al., 3 Jul 2025, O'Donnell et al., 2023, Harrow et al., 2018).