Approximate Circuit Complexity

• Approximate circuit complexity is a family of resource measures that relax exact synthesis by employing error thresholds in computing Boolean functions, unitaries, or quantum states.
• It encompasses various paradigms, including multiplicative complexity, polynomial approximation, and block-encoding, each with distinct model-dependent error metrics.
• Applications range from cryptographic hardness and quantum compression to field-theoretic and AQEC lower bounds, highlighting diverse methodological challenges.

Approximate circuit complexity is a family of resource measures in which the cost of computing a Boolean function, synthesizing a unitary, or preparing a quantum state is optimized subject to an approximation criterion rather than exact realization. Across current literature, the approximation relation may be multiplicative on a classical complexity measure, pointwise additive error in polynomial approximation, norm-bounded error for operator synthesis, fidelity or energy-constrained channel error for quantum implementations, or restriction to a tractable state-and-gate manifold such as Gaussian states (Find, 2014, Bun et al., 2013, Camps et al., 2020, Shou et al., 15 Apr 2026, Yi et al., 2023). Taken together, these works show that approximate circuit complexity is not a single invariant but a collection of model-dependent notions whose technical content depends on the choice of gate set, ambient geometry, error metric, and computational representation.

1. Foundational approximation paradigms

A basic distinction is between exact circuit complexity and its approximate analogues. For an $n$-qubit unitary $U$, exact circuit complexity can be defined as

$$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$

whereas a standard approximate version is

$$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$

(Haferkamp et al., 2021). In the Boolean setting, approximation may instead mean producing a value within a multiplicative factor of an exact measure, as in approximation of multiplicative complexity from a truth table (Find, 2014). In polynomial-approximation settings, the resource is degree rather than gate count, and approximation is pointwise sup-norm error $\|p-f\|_\infty\le \epsilon$ (Bun et al., 2013).

Quantum synthesis introduces further variants. In the block-encoding framework, a unitary $U_n$ on $n=a+s$ qubits is an $(\alpha,a,\epsilon)$-block-encoding of $A_s$ if

$$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$

so approximation is transferred from exact synthesis of $U$0 to an approximate ancilla-assisted embedding (Camps et al., 2020). In passive linear optics, robust circuit complexity asks for the minimum number of gates required to realize some $U$1 such that $U$2 is small, with the Hilbert–Schmidt error subsequently lifted to high-fidelity guarantees for low-energy bosonic states (Shou et al., 15 Apr 2026).

A different strand defines approximation through robustness of state preparation. For an $U$3 code, the $U$4-robust all-to-all circuit complexity $U$5 and its geometrically local analogue $U$6 are lower-bounded in terms of subsystem variance and code rate $U$7, with the operative criterion

$U$8

(Yi et al., 2023). A thermodynamic formulation goes ఇంకా further by defining approximate counting of circuits within an $U$9-ball around a target unitary,

$$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$0

and the associated approximate complexity

$$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$1

(Chamon et al., 2023).

These formulations are not equivalent. This suggests that “approximate circuit complexity” is best understood as a meta-category of optimization problems rather than a single canonical complexity measure.

2. Boolean functions, multiplicative complexity, and polynomial approximation

In Boolean circuit complexity, one exact measure that admits a nontrivial approximation theory is multiplicative complexity. For $$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$2, working over XOR-AND circuits on the basis $$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$3 with fanin $$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$4, the multiplicative complexity is

$$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$5

(Find, 2014). Unlike ordinary circuit size, this counts only AND gates and treats XOR as free. The same paper contrasts this with nonlinearity

$$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$6

which is the Hamming distance to the nearest affine function.

The computational contrast is sharp. From a truth table, nonlinearity is computable in $$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$7 time by the Fast Walsh Transform, while multiplicative complexity is cryptographically hard: under the existence of pseudorandom function families, hence under one-way functions, no $$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$8-time algorithm computes $$C(U):=\min\{\text{number of 2-qubit gates in any circuit that exactly implements }U\},$$9 from the truth table, and for every constant $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$0, no such algorithm approximates it within factor $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$1 (Find, 2014). For circuit input, the decision problem

$$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$2

lies in $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$3, while the same approximation barrier persists for algorithms running in $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$4 time.

A second major approximation framework replaces circuits by low-degree polynomials. The $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$5-approximate degree $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$6 is the minimum degree of a real polynomial $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$7 with $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$8, and the one-sided version imposes asymmetric constraints on the $$C(U,\eta):=\min\{\text{\# of 2-local gates implementing }V\text{ such that }\|U-V\|\le \eta\}$$9 inputs of $\|p-f\|_\infty\le \epsilon$0 (Bun et al., 2013). A generic hardness-amplification theorem shows that if $\|p-f\|_\infty\le \epsilon$1, then for

$\|p-f\|_\infty\le \epsilon$2

on disjoint copies, one has $\|p-f\|_\infty\le \epsilon$3 (Bun et al., 2013). Thus OR-composition can amplify constant-error one-sided hardness into near-maximal-error hardness.

This mechanism yields explicit lower bounds for constant-depth circuits. There exists a polynomial-size depth-three circuit $\|p-f\|_\infty\le \epsilon$4 such that any degree-$\|p-f\|_\infty\le \epsilon$5 polynomial cannot pointwise approximate $\|p-f\|_\infty\le \epsilon$6 to error better than

$\|p-f\|_\infty\le \epsilon$7

(Bun et al., 2013). The same techniques give

$\|p-f\|_\infty\le \epsilon$8

for depth-$\|p-f\|_\infty\le \epsilon$9 AND-OR trees, tight up to polylogarithmic factors for constant $U_n$0 (Bun et al., 2013). In this literature, approximate circuit complexity is mediated by a surrogate measure—polynomial degree—that controls communication complexity, threshold weight, and learning-theoretic hardness.

3. Approximate quantum synthesis and compressibility

Approximate synthesis problems ask when a large operator can be implemented by substantially fewer gates than exact generic synthesis would require. The block-encoding framework provides one structured answer. If an operator $U_n$1 admits a canonical polyadic-like decomposition

$U_n$2

with $U_n$3, efficient block-encodings for each factor, and efficient coefficient-state preparation, then the full operator can be block-encoded with $U_n$4 gate complexity, hence polylogarithmic complexity in the matrix dimension $U_n$5 (Camps et al., 2020). The framework composes smaller block-encodings by Kronecker products and LCU-style linear combinations, shifting the synthesis problem from exact unitary realization to structured approximation.

Random linear optics exhibits a different phenomenon: compressibility of typical finite-depth circuits. For a depth-$U_n$6 random one-dimensional brickwall passive linear optical unitary $U_n$7, with probability $U_n$8, there exists a unitary $U_n$9 built from only

$n=a+s$0

nearest-neighbor gates such that

$n=a+s$1

(Shou et al., 15 Apr 2026). Since the exact circuit uses $n=a+s$2 gates, the robust circuit complexity grows diffusively rather than ballistically in depth. The associated Gaussian unitary $n=a+s$3 also satisfies

$n=a+s$4

so the reduced gate count has an operational interpretation on bounded-energy inputs (Shou et al., 15 Apr 2026).

Approximation in shallow quantum circuit classes may also be highly constrained. In QAC circuits—constant-depth quantum analogues of AC$n=a+s$5 with one-qubit gates and generalized Toffoli gates—there are depth-$n=a+s$6 upper bounds for approximating parity of size

$n=a+s$7

for all $n=a+s$8, yet arbitrary depth-$n=a+s$9 QAC circuits require at least $(\alpha,a,\epsilon)$0 multi-qubit gates to achieve a $(\alpha,a,\epsilon)$1 approximation, and depth-2 QAC circuits cannot achieve $(\alpha,a,\epsilon)$2 approximation of parity, fanout, restricted fanout, or cat-state preparation (Rosenthal, 2020). Approximate complexity in this setting is therefore governed simultaneously by depth, size, and the nonclassical structure of allowed multi-qubit layers.

A recurring limitation appears in random-circuit lower bounds. The proof that exact complexity of Haar-random two-qubit circuits grows linearly for exponentially long times does not directly extend to standard approximate complexity, because thickening the low-dimensional image of an architecture by an $(\alpha,a,\epsilon)$3-neighborhood can destroy the dimension-gap obstruction (Haferkamp et al., 2021). The same work proves only a slightly robust statement: for sufficiently small $(\alpha,a,\epsilon)$4, every low-depth circuit $(\alpha,a,\epsilon)$5 remains at a positive Frobenius distance from the random-circuit unitary with high probability. This delineates a methodological gap between exact and approximate lower bounds.

4. Geometric and field-theoretic notions of approximate complexity

In quantum field theory, “approximate circuit complexity” often means restricting the admissible states and gates to a tractable submanifold. For the free $(\alpha,a,\epsilon)$6 gauge field in Coulomb gauge, the relevant sector is the manifold of Gaussian states connected by Bogoliubov transformations. Complexity is computed via Nielsen’s geometric approach as the length of the shortest path on the Lie group of Bogoliubov transformations equipped with a right-invariant metric (Moghimnejad et al., 2021). If $(\alpha,a,\epsilon)$7 and $(\alpha,a,\epsilon)$8 are the reference and target covariance matrices and $(\alpha,a,\epsilon)$9, then

$A_s$0

For the free vector field in $A_s$1 dimensions, the ground-state complexity becomes

$A_s$2

and in the continuum limit scales as

$A_s$3

(Moghimnejad et al., 2021). The restriction to Gaussian states and symplectic transformations is explicitly an approximation to the full field-theoretic problem.

A related analysis in Proca theory uses both Nielsen and Fubini–Study approaches. After lattice regularization and mode decomposition, the approximate ground state is generated from a reference Gaussian by momentum-dependent squeezing, leading in the FS formulation to

$A_s$4

(Meng et al., 2021). For thermofield-double states, the two approaches diverge dynamically: Nielsen’s method gives a complexity that behaves approximately as $A_s$5 after subtracting the $A_s$6 value, whereas the FS metric gives linear growth $A_s$7 (Meng et al., 2021). This demonstrates that approximate complexity in QFT is sensitive to the chosen geometric formalism even when the state family is fixed.

A categorical variant appears in two-dimensional TQFT. In the bordism category $A_s$8, exact complexity is the minimal pants decomposition count, but after applying the TQFT functor into Hilbert space exact universality typically fails. Approximation then becomes essential: in semisimple theories, repeated application of the handle operator can generate dense orbits on a torus of states, and complexity is interpreted as the minimal number of handle applications needed to approximate a target state to tolerance $A_s$9 (Couch et al., 2021). Here approximate circuit complexity is neither gate-synthesis error nor operator norm distance on a unitary group; it is a discrete orbit-hitting problem on a sparse image of Euclidean path integrals.

These constructions share a common pattern. A large or infinite-dimensional synthesis problem is projected onto a structured manifold—Gaussian symplectic or orthogonal transformations, handle-operator dynamics, or other restricted sectors—where geodesic or combinatorial optimization becomes feasible. This suggests that much of field-theoretic approximate circuit complexity is best interpreted as a controlled proxy rather than a direct extension of finite-gate exact complexity.

5. Complexity lower bounds from AQEC, local indistinguishability, and many-body order

Approximate quantum error correction provides a robust route to lower bounds on circuit complexity. For an $$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$0 code $$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$1, the subsystem variance

$$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$2

quantifies local distinguishability of code states (Yi et al., 2023). If

$$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$3

then every code state obeys the all-to-all lower bound

$$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$4

and for a $$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$5-dimensional lattice graph $$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$6,

$$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$7

(Yi et al., 2023). The threshold is naturally described as $$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$8, or roughly $$\tilde A_s=\left(\bra{0}^{\otimes a}\otimes I_s\right)U_n\left(\ket{0}^{\otimes a}\otimes I_s\right),\qquad \|A_s-\alpha \tilde A_s\|_2\le \epsilon,$$9 when $U$00. In this regime, sufficiently good AQEC forces all code states to be circuit-complex.

A complementary 2025 result strengthens the local-indistinguishability mechanism by combining clustering assumptions with a generalized Lovász local lemma. If two states $U$01 satisfy $U$02 and both have circuit complexity at most $U$03, then there exists an operator $U$04 with $U$05 and support size at most $U$06 such that

$U$07

(Yi et al., 6 Oct 2025). Orthogonal short-range entangled states therefore cannot remain locally indistinguishable. Applied to AQEC codes with universal transversal gates or transversal Clifford gates, this yields lower bounds on approximate circuit complexity for every code state; applied to $U$08, it gives $U$09 for one-dimensional local circuits and $U$10 for all-to-all circuits in the regimes stated in the paper (Yi et al., 6 Oct 2025).

These AQEC-based results connect approximate complexity to long-range entanglement, topological order, and criticality. Exact $U$11 codes are recovered as the $U$12 special case, while approximate codes interpolate between strict local indistinguishability and complexity lower bounds. The common principle is that if logical information is hidden too well from local probes, then shallow preparation becomes impossible.

6. Adjacent optimization problems and broader structural themes

Several recent works study optimization problems that are not themselves standard approximate complexity measures but are closely adjacent to the subject. One example is the circuit width problem for Montanaro’s polynomial representation of quantum circuits over $U$13. For a degree-three polynomial $U$14 with no constant term,

$U$15

and deciding whether $U$16 is NP-complete (Ji et al., 16 Jun 2026). Approximation is also hard: for every $U$17, it is NP-hard to distinguish

$U$18

even for degree-two polynomials (Ji et al., 16 Jun 2026). Since efficient width minimization would provide an approximate-counting route for normalized polynomial gaps, this parameter links structural circuit optimization to approximate simulation.

Another example is quantum circuit cutting. Representing a circuit as a legal DAG, wire cuts are modeled by edge duplication, and the optimization problem asks for a set of duplicated edges that partitions the circuit into acceptable bounded-size clusters while minimizing the number of cuts (Idan et al., 26 Apr 2026). The resulting Graph Duplication problem is NP-complete in general, NP-complete even for 2-legal DAGs corresponding to one- and two-qubit gate sets, and admits only exact exponential-time methods such as an SMT formulation for bounded instances (Idan et al., 26 Apr 2026). Because wire cuts induce classical overhead scaling as $U$19 for $U$20 cut edges in the discussed framework, optimal cut placement is an approximate-execution resource tradeoff rather than a pure exact-synthesis problem.

A broader statistical interpretation is developed in the thermodynamic treatment of circuit functionality. There, for fixed functionality $U$21 and size $U$22, the circuit entropy is

$U$23

while for approximate quantum circuits the count $U$24 measures the number of length-$U$25 gate sequences within distance $U$26 of $U$27 (Chamon et al., 2023). The induced complexity distribution over fixed-size circuits is sharply peaked at a typical $U$28, corresponding to a finite compression factor

$U$29

(Chamon et al., 2023). At the same time, the paper argues that the space of same-size, same-function circuits generically fragments into disconnected ergodic sectors under local functionality-preserving moves, and that a universal polynomial-length connectivity witness would imply $U$30 and collapse the polynomial hierarchy (Chamon et al., 2023). Approximate circuit complexity therefore has a second, coarse-grained meaning: not merely distance to a target object, but compressibility and mixing within constrained sectors of circuit space.

A common misconception is that approximation invariably makes circuit optimization easy. The literature surveyed here shows the opposite. Multiplicative complexity remains exponentially inapproximable from truth tables under standard cryptographic assumptions (Find, 2014); circuit width is NP-hard to approximate within a nontrivial constant factor (Ji et al., 16 Jun 2026); optimal circuit cutting is NP-complete even on heavily restricted gate models (Idan et al., 26 Apr 2026); and random or highly ordered quantum states can remain provably resistant to shallow approximate preparation (Yi et al., 6 Oct 2025, Yi et al., 2023). Approximation changes the question, but it does not uniformly soften it.

Approximate circuit complexity is therefore best viewed as a structured landscape rather than a single doctrine. In some regimes it exposes compressibility, as in random linear optics (Shou et al., 15 Apr 2026); in others it sharpens hardness, as in one-sided approximate degree and multiplicative complexity (Bun et al., 2013, Find, 2014); and in field theory or many-body physics it often serves as a controlled proxy tied to geometry, symmetry, or local indistinguishability (Moghimnejad et al., 2021, Yi et al., 2023). The unifying theme is resource minimization under relaxation, but the mathematics depends decisively on what is being approximated, by which circuits, and in what metric.