Spectral Forrelation in Quantum Complexity

• Spectral Forrelation is a promise problem defined via the squared operator norm of the restricted Hadamard transform between subsets of the Boolean hypercube.
• It measures two-basis compatibility by determining if a quantum state can be well supported in both the standard and Hadamard bases.
• Its complexity framework leverages oracle access and singular value analysis to exhibit promise gaps that separate QMA quantum witnesses from classical QCMA proofs.

Spectral Forrelation is a promise problem on two subsets $S,U\subseteq\{0,1\}^n$ of the Boolean hypercube, defined by the squared operator norm of the Hadamard transform restricted from the coordinate subspace supported on $S$ to the coordinate subspace supported on $U$. Equivalently, it asks whether there exists a quantum state whose standard-basis measurement distribution is well supported on $S$ and whose Fourier, or Hadamard-basis, measurement distribution is well supported on $U$. In the complexity-theoretic setting of oracle access, this quantity becomes the central decision problem used to separate $\mathrm{QMA}$ from $\mathrm{QCMA}$ relative to a classical oracle (Bostanci et al., 12 Nov 2025).

1. Formal definition

Let

$$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$

where $\Pi_S$ and $\Pi_U$ are the projectors onto the subspaces spanned by standard basis states supported on $S$0 and $S$1, respectively, and let $S$2 denote the $S$3-qubit Hadamard transform. The pair $S$4 is $S$5-spectrally Forrelated when

$S$6

The matrix

$S$7

is the relevant “Forrelation matrix.” Spectral Forrelation is therefore a property of a pair of subsets, not of a single Boolean function, and its value is the squared spectral norm of this restricted Hadamard operator.

The paper also gives an operational interpretation. If $S$8, then after applying $S$9 and measuring in the standard basis, the probability mass landing in $U$0 is

$U$1

In this formulation, spectral Forrelation is a “two-basis” compatibility condition: the same state must be simultaneously compatible with support constraints in the standard basis and in the Hadamard basis (Bostanci et al., 12 Nov 2025).

2. Restricted Hadamard structure and singular-value viewpoint

The defining quantity can be read directly as the largest singular value squared of a restricted Hadamard submatrix. If the Boolean hypercube basis is indexed by $U$2, then

$U$3

Hence $U$4 is a submatrix of the Walsh–Hadamard matrix, and spectral Forrelation asks whether its top singular value is large.

This leads to the yes/no structure used in the oracle problem. A yes instance is one for which there exists a witness state $U$5 such that

$U$6

is large; a no instance is one for which every $U$7 yields small overlap. The paper states a representative promise gap as follows: yes if the spectral Forrelation is at least $U$8, and no if it is at most $U$9. More generally, for any $S$0, the verifier complexity scales like

$S$1

queries via Marriott–Watrous amplification (Bostanci et al., 12 Nov 2025).

A common conflation is to identify this quantity with standard scalar Forrelation. The operator-norm formulation is materially different: instead of a single Fourier-correlation scalar, one asks for a large singular direction of a restricted transform. Standard Forrelation is described in the paper as a one-dimensional, rank-1-like analogue.

3. Oracle access, promise formulation, and QMA verification

The sets $S$2 and $S$3 are treated as oracle-accessible Boolean functions

$S$4

with $S$5 iff $S$6, and likewise for $S$7. Access is by membership queries, implemented as phase oracles: $S$8

The decision problem is a promise problem on oracle pairs:

• yes: $S$9 is at least $U$0-spectrally Forrelated;
• no: $U$1 is at most $U$2-spectrally Forrelated.

The formulation extends naturally to multisets by taking support as a set, and to Boolean functions by interpreting the $U$3-preimage as the set. This keeps the notion compatible with standard oracle language while preserving the geometric meaning of the projectors.

The same structure gives a direct $U$4 verification procedure. A quantum witness can be chosen as a top singular vector of $U$5. The verifier checks support in $U$6, applies the Hadamard transform, checks support in $U$7, and accepts with probability

$U$8

Thus spectral Forrelation lies naturally in $U$9: the witness is not a classical string naming a special point, but a structured quantum state encoding the top singular direction (Bostanci et al., 12 Nov 2025).

4. Complexity-theoretic role in the QMA–QCMA oracle separation

The principal use of spectral Forrelation is to build a classical oracle relative to which

$\mathrm{QMA}$0

Equivalently, there is a language $\mathrm{QMA}$1 decidable by a polynomial-time quantum verifier with a quantum witness but not by any polynomial-time quantum verifier that is restricted to a classical witness.

The distinction exploited in the lower-bound strategy is that a classical witness can be copied and reused, whereas a quantum witness is a “use once” object. The paper identifies this as the key observation behind the separation. If a $\mathrm{QMA}$2 verifier for spectral Forrelation existed, then repeated executions with the same classical advice could be converted into a sampling procedure. This does not simultaneously yield a $\mathrm{QMA}$3 lower bound, precisely because the quantum witness lacks the same reusability.

The reduction is organized in three steps. First, assume a $\mathrm{QMA}$4 algorithm for spectral Forrelation. Second, convert that verifier into a sampler that, given only access to $\mathrm{QMA}$5, can produce many points from $\mathrm{QMA}$6. Third, show that such a sampler would have to succeed with probability much larger than is actually possible. The contradiction gives the $\mathrm{QMA}$7 lower bound and therefore the relativized separation (Bostanci et al., 12 Nov 2025).

This role explains why spectral Forrelation is formulated as a set-based operator problem rather than as a scalar Fourier correlation. Its geometry is rich enough to admit a natural quantum witness and rigid enough to support a sampling-hardness argument against classical witnesses.

5. Strong instances, sampling hardness, and the bosonic compressed-oracle method

The reduction from verification to sampling is formalized using strong yes instances. A pair $\mathrm{QMA}$8 is strong if it is a yes instance and every small $\mathrm{QMA}$9 makes $\mathrm{QCMA}$0 a no instance. This rules out a trivial classical strategy in which the witness merely lists a few favorable points of $\mathrm{QCMA}$1.

For the hard distribution, the paper defines $\mathrm{QCMA}$2 over pairs $\mathrm{QCMA}$3 by first choosing $\mathrm{QCMA}$4 as a sparse random multiset of size

$\mathrm{QCMA}$5

for a small constant $\mathrm{QCMA}$6, then setting

$\mathrm{QCMA}$7

and finally including each $\mathrm{QCMA}$8 with probability

$\mathrm{QCMA}$9

for a small constant $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$0. The paper proves that with overwhelming probability, $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$1 is a strong yes instance.

The hardest part is then to upper bound how well a quantum algorithm can sample from $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$2 using only $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$3. For this, the paper introduces a bosonic “second quantization” perspective on compressed oracle techniques. The multiset $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$4 is encoded as a bosonic state, and a random multiset of size $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$5 corresponds to

$$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$6

Queries to $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$7 are expressed through operators built from a “double hopping” operator

$$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$8

Within this compressed picture, the analysis shows that after a polynomial number of queries, the algorithm’s purified state remains close to a quasi-even condensate: most bosons stay in the zero-momentum mode, and only few momentum modes have odd occupancy. Such states have low probability of producing many distinct position samples from $$\Pi_S=\sum_{x\in S}\lvert x\rangle\langle x\rvert,\qquad \Pi_U=\sum_{x\in U}\lvert x\rangle\langle x\rvert,$$9. Therefore the sampler promised by the $\Pi_S$0 reduction cannot exist, completing the lower bound (Bostanci et al., 12 Nov 2025).

6. Relation to other Forrelation frameworks

The broader Forrelation literature contains several mathematically distinct objects that share a Fourier-correlation motif but differ substantially in domain, codomain, and complexity role.

Variant	Underlying data	Central quantity
Standard Forrelation	Boolean functions $\Pi_S$1 or $\Pi_S$2	Scalar $\Pi_S$3
Spectral Forrelation	Sets $\Pi_S$4	$\Pi_S$5
$\Pi_S$6-Forrelation	Boolean functions with $\Pi_S$7-Hadamard layers	$\Pi_S$8

In the Boolean-function setting, 2-fold and 3-fold Forrelation are scalar quantities computed from phase-oracle access to functions $\Pi_S$9. These formulations are used to study Walsh spectrum sampling, cross-correlation, and autocorrelation, including constant-query cross-correlation sampling and direct estimation of $\Pi_U$0 through 3-fold Forrelation circuits (Dutta et al., 2021). A later generalization replaces the fixed Walsh–Hadamard phase structure with an $\Pi_U$1-dependent transform $\Pi_U$2, defines $\Pi_U$3-Forrelation, and shows how the framework subsumes the Walsh-Hadamard, nega-Hadamard, and $\Pi_U$4-Hadamard settings while preserving constant-query estimation and sampling procedures (Dutta et al., 9 Jul 2025).

Another distinct branch studies Forrelation as a constraint atom in random sparse CSPs. In that setting, the $\Pi_U$5-bit Forrelation function is analyzed as a two-eigenvalue $\Pi_U$6CSP, and the SDP value, quantum value, and spectral relaxation value coincide asymptotically for random $\Pi_U$7-regular instances, yielding

$\Pi_U$8

with high probability (Mohanty et al., 2019).

These neighboring uses clarify the specific place of spectral Forrelation. It is neither the scalar Boolean-function quantity used in oracle algorithms nor the CSP atom used in random-instance spectral analysis. Rather, it is a set-based restricted-Hadamard norm problem designed to expose a genuine gap between quantum and classical witnesses. A plausible implication is that the shared Fourier-analytic vocabulary masks substantial differences in witness structure, oracle access model, and lower-bound methodology across the various Forrelation frameworks.