Quantum-Classical Separation

Updated 4 December 2025

Quantum-classical separation is the rigorous demonstration that quantum systems outperform classical methods in efficiency, resource use, or computational guarantees.
Key frameworks involve complexity and communication models, where quantum advantages manifest as exponential or even unbounded gaps compared to classical counterparts.
This concept underpins quantum supremacy, affecting fields from secure communication to optimization, and raises open challenges in algorithm design and experimental validation.

Quantum-classical separation denotes the rigorous demonstration that certain computational, informational, or resource-bounded tasks can be accomplished more efficiently, with fewer resources, or with fundamentally stronger guarantees by quantum systems than by any classical system considered within the analogous constraints. This concept is central to quantum information theory, computational complexity, communication complexity, learning theory, and quantum foundations. Quantum-classical separation can refer to unbounded (often exponential or infinite) gaps in complexity, success probability, communication cost, or available information, and may be unconditional (provable with no unproven assumptions) or conditional (based on plausible, often cryptographic, hypotheses).

1. Complexity-Theoretic Foundations and Main Formalisms

Quantum-classical separation is fundamentally encoded in differences between classical and quantum computational (and communication) complexity classes and related models of information processing. Core classes and relations include:

BPP (Bounded-error Probabilistic Polynomial time): Classical randomized decision problems solvable with error ≤ 1/3 in poly(n) time.
BQP (Bounded-error Quantum Polynomial time): Quantum analog, allowing uniform poly-size quantum circuits.
FBPP/FBQP: Functional variants outputting a valid witness y for input x, satisfying a poly-time checkable relation R(x, y), with bounded error.
PostBPP/PostBQP: Classical/quantum circuits with “postselection”—condition on a low-probability event b=1, and then solve the problem over this filtered distribution.
PH (Polynomial Hierarchy): Generalization of NP coNP, ΣkP = NPk-1P, etc., layered alternations of existential and universal quantifiers.
#P (Counting class): Classical class counting the number of accepting paths of a nondeterministic poly-time Turing machine.
Approximate Counting Oracle: Returns an estimate $\hat{p}$ such that $(\#P(N, x))/(1+\varepsilon)\leq \hat{p} \leq (1+\varepsilon)\#P(N, x)$ .
SampBQP/SampP: Classes of sampling problems efficiently solvable by quantum/classical poly-time (up to total variation distance error).
Random reducibility/self-reducibility: A property or reduction where a function is approximated by randomized queries to an oracle, marginally independent of input.
Communication/information complexity setup: Cost of solving distributed tasks over one-way, two-way, or multiparty (e.g., number-on-forehead) protocols, comparing quantum vs. classical resources.

This framework supports formal statements about quantum-classical separations, including exact theorems and constructions (Marshall et al., 28 Oct 2024, Perry et al., 2014, Montanaro, 2010, Anshu et al., 2016, Kretschmer et al., 8 Sep 2025, Yang et al., 20 Jun 2025).

2. Exponential and Infinite Separations: Proof Techniques and Exemplars

Provable quantum-classical separation manifests in various ways, often via exponential or even infinite gaps:

Sampling and decision tasks: If classical approximate simulation of certain quantum sampling experiments (e.g., BosonSampling, IQP, DQC1) were possible, the polynomial hierarchy PH would collapse to the second level ( $\Sigma^P_2$ $Σ_{2}^{P}$ ), considered highly implausible. Therefore, unless PH collapses, quantum computers can sample from distributions (even up to additive error in total variation) that classical computers cannot, establishing $SampBQP\neq SampP$ $S am pBQP \neq = S am pP$ and $FBQP\neq FBPP$ $FBQP \neq = FBPP$ under natural assumptions (Marshall et al., 28 Oct 2024).
- Key theorem: If $P^{\#P} = BPP^{NP}_{\parallel}$ , then $PH$ collapses to $\Sigma^P_2 \cap \Pi^P_2$ . If quantum and classical postselection are equally powerful ( $PostBQP=PostBPP$ ), $PH$ also collapses to second level.
- Hardness of sampling: Additive-error classical simulation of BosonSampling, IQP, or DQC1 distributions implies $FBQP=FBPP$ unless PH collapses (Marshall et al., 28 Oct 2024).
Unconditional information-theoretic separation: There exist explicit protocols where a quantum message of n qubits encodes a task that any classical protocol requires Ω(2ⁿ⁾ bits to match the performance. The cross-entropy benchmark task—implemented on a trapped ion device—demonstrates that a 12-qubit state cannot be simulated by any classical message under 62 bits for the achieved fidelity, rising to approximately 2ⁿ bits in the limit (Kretschmer et al., 8 Sep 2025). This is unconditional and does not rely on complexity conjectures, marking a milestone in practical quantum advantage.
Quantum-classical separation in communication complexity: For tasks like the exclusion game, any classical protocol solving the task exactly must transmit Ω(n) bits, while a quantum protocol achieves zero error with vanishing information cost as n→∞ (an infinite separation). Even with entanglement assistance, classical protocols need Ω(n) bits, while a quantum protocol requires only O(1) classical bits with nonzero abort probability (Perry et al., 2014).
Functional, circuit, and communication separations: PM-INVARIANCE and related promise communication tasks have one-way quantum communication of O(log n) qubits but require 2^{{Ω(n^{7/16})}} bits classically (an exponential gap). In three-party number-on-forehead (NOF) communication, certain gadgeted hidden matching problems have quantum protocols with O(log n) cost and randomized classical protocols with Ω(n^{1/16}) lower bounds, representing the first exponential separation in a NOF regime (Montanaro, 2010, Yang et al., 20 Jun 2025).

A table summarizing some representative separations:

Task/Class	Quantum Cost	Classical Cost	Nature of Separation
BosonSampling/IQP/DQC1 sampling (Marshall et al., 28 Oct 2024)	Poly-time (quantum)	Not poly-time unless PH collapses	Conditional exponential
Cross-entropy (DXHOG) (Kretschmer et al., 8 Sep 2025)	n qubits	≥Ω(2ⁿ⁾ bits	Unconditional exponential
Exclusion game (Perry et al., 2014)	0 (quantum info cost, asymptotically)	Ω(n) (classical info cost)	Infinite/unbounded
Communication (PM-INVARIANCE) (Montanaro, 2010)	O(log n) qubits	2^{{Ω(n^{7/16})}} bits	Exponential one-way
Three-NOF GHM (Yang et al., 20 Jun 2025)	O(log n) qubits	Ω(n^{1/16}) bits	Superpolynomial multiparty

3. Quantum-Classical Separation in Learning and Optimization

Quantum-classical separation also appears in learning theory and continuous optimization:

Density modeling: Under standard cryptographic assumptions (weak pseudo-random functions built from Diffie–Hellman), there exist distribution classes for which a quantum learner can construct an (ε, δ)-PAC evaluator in polynomial time (due to efficient quantum discrete-log), but no classical poly-time evaluator exists. This is a super-polynomial separation; the proof methodology leverages hardness of distinguishing PRF samples from random, which is reducible to the existence of classical evaluator models in the PAC sense (Pirnay et al., 2022).
PAC learning separations: Systematic frameworks ("checklists") clarify that cryptography-based hardness is required for separations in the PAC model with classical data. If function inversion is classically heuristically hard but efficiently invertible quantumly (e.g., discrete log, factoring), and if learning the residual target is quantum-efficient, then a quantum-classical gap results. Without cryptographic input, no "natural" separations are known (Gyurik et al., 2022).
Optimization: For carefully designed families of $d$ -dimensional smooth nonconvex functions with $2^d$ local minima, quantum Hamiltonian descent (adiabatic, continuous-time quantum) finds a global minimizer in $\tilde{O}(d^3)$ queries and $\tilde{O}(d^4)$ gates (polynomial in $d$ ), whereas all leading classical algorithms scale exponentially in $d$ . No polynomial-time classical algorithm was found for these instances (Leng et al., 2023).

4. Oracle and Resource-Bounded Separations

Oracle separations rigorously establish that certain resource models (quantum circuits/classical-quantum hybrids) are strictly more powerful than classical models—sometimes optimally so:

Optimal quantum depth separation: For any d, there exists an oracle such that increasing quantum circuit depth from $d$ to $d+1$ strictly increases the class of decision problems (even when arbitrary polynomial-time classical processing is allowed between quantum layers). The "d-Bijective Shuffling Simon’s Problem" concretely realizes this gap (Hasegawa et al., 2022).
Incomparability of quantum-classical hybrids: Oracle constructions show that circuits with poly-many shallow quantum subroutine calls (CQ_d) and those with shallow quantum layers interleaved with classical computation (QC_d) are generally incomparable: each can solve tasks the other cannot under appropriate oracles, and both are strictly weaker than full BQP (Arora et al., 2022).

These results highlight that even unbounded classical computation cannot compensate for limited quantum resources beyond sharp thresholds.

5. Quantum-Classical Separation in Foundations and Measurement

Quantum-classical separation is also formalized in quantum foundations:

Measurement contextuality: Quantum systems exhibit correlations (e.g., in the magic square, GHZ, or hidden linear function games) that cannot be accounted for by any noncontextual hidden-variable theory. Carefully designed tasks achieve success probabilities, violations of inequalities, or resource savings (qubit number, circuit depth) that no classical strategy can match. Such experiments function as practical benchmarks of quantum contextuality and computational power of quantum hardware (Kumar et al., 1 Dec 2025).
Decomposition of uncertainty: Quantum measurement uncertainty is formally separated into classical and quantum parts using Kirkwood–Dirac quasiprobability and generalized entropy. The "genuine quantum part" vanishes if and only if state and measurement commute; its positivity is both necessary and sufficient for contextuality proofs via weak measurement/postselection (Budiyono, 13 Dec 2024).
Probability ratio tests: For three binary measurements, the ratio of pairwise "same" to "different" probabilities is 1/2 classically (assuming no three-way coincidences), but can be brought to 1/3 in quantum maximally entangled states; for general entanglement $|\psi(r)\rangle$ , the threshold $r<5.83$ certifies quantum-classical separation. A value of 5/12 is a practical threshold under realistic noise (Kak, 2013).

6. Significance, Implications, and Open Problems

Quantum-classical separation directly underpins claims of quantum advantage in computation, simulation, communication, and learning. Demonstrations range from communication and information-theoretic tasks with infinite gaps (Perry et al., 2014, Kretschmer et al., 8 Sep 2025), to physically plausible computational tasks resistant to classical simulation (Marshall et al., 28 Oct 2024, Leng et al., 2023, Grønlund et al., 4 Nov 2024), to practical algorithms validated on scalable hardware (Kretschmer et al., 8 Sep 2025, Kumar et al., 1 Dec 2025).

Critical implications include:

Quantum supremacy—strong evidence that quantum hardware accesses computational regimes inaccessible to classical machines of similar physical or resource scale.
Limits of classical simulation—classical computers require exponentially more resources (time, memory, communication) to replicate certain quantum protocols, even with oracular or sampling capabilities.
Polynomial hierarchy—conditional proof techniques show that approximation of quantum output distributions by classical samplers would cause implausible collapses in classical complexity hierarchy.
Unconditional resource-based supremacy—quantum devices can solve explicitly constructed tasks using n qubits, while any classical device needs Ω(2ⁿ⁾ bits, even for modest fidelity.
Learning and optimization—quantum devices can learn or optimize in settings where classical algorithms are provably inefficient under widely believed cryptographic assumptions or via explicit constructions.
Contextuality-based benchmarks—quantum hardware can robustly demonstrate nonclassicality via violation of contextuality inequalities or efficient solution of classically hard problems under bounded resources.

Open questions include finding oracle-free (unrelativized) separations for broader classes (beyond quantum Fourier sampling and error-corrected cryptographic primitives), characterizing the minimal required quantum resources for robust supremacy, developing practical tasks for near-term noisy devices, and extending foundational separations (e.g., via contextuality measures) to inform experimental benchmarks and device certification.

References (key arXiv IDs):

(Marshall et al., 28 Oct 2024): Improved separation between quantum and classical computers for sampling and functional tasks.
(Leng et al., 2023): A quantum-classical performance separation in nonconvex optimization.
(Grønlund et al., 4 Nov 2024): An Exponential Separation Between Quantum and Quantum-Inspired Classical Algorithms for Linear Systems.
(Perry et al., 2014): Communication tasks with infinite quantum-classical separation.
(Montanaro, 2010): A new exponential separation between quantum and classical one-way communication complexity.
(Kretschmer et al., 8 Sep 2025): Demonstrating an unconditional separation between quantum and classical information resources.
(Yang et al., 20 Jun 2025): Quantum versus Classical Separation in Simultaneous Number-on-Forehead Communication.
(Pirnay et al., 2022): A super-polynomial quantum-classical separation for density modelling.
(Anshu et al., 2016): Exponential Separation of Quantum Communication and Classical Information.
(Kak, 2013): Probability and the Classical/Quantum Divide.
(Kumar et al., 1 Dec 2025): Quantum-Classical Separation in Bounded-Resource Tasks Arising from Measurement Contextuality.
(Budiyono, 13 Dec 2024): Separation of measurement uncertainty into quantum and classical parts based on Kirkwood-Dirac quasiprobability and generalized entropy.
(Gyurik et al., 2022): On establishing learning separations between classical and quantum machine learning with classical data.
(Arora et al., 2022): Oracle separations of hybrid quantum-classical circuits.
(Hasegawa et al., 2022): An optimal oracle separation of classical and quantum hybrid schemes.