Quantum Instance Complexity

• Quantum Instance Complexity is a per-instance measure capturing the size of quantum solver descriptions, copy complexity in state certification, and tree-size expression of quantum states.
• It distinguishes quantum from classical resources by evaluating instance-specific gaps, such as queasiness factors and exponential utility across inputs.
• This framework underpins applications in quantum algorithm design, SAT instance reductions, and secure zero-knowledge proofs in quantum settings.

Quantum instance complexity denotes a family of instance-by-instance complexity notions in quantum information and quantum computation. In current usage, the “instance” may be a fixed classical input $x$ to a language $L$, a target density operator $\sigma$ in a certification problem, or a quantum state $\rho$ supplied directly as the input of a promise problem. Correspondingly, the quantity of interest may be the size of the shortest quantum solver description, the copy complexity of certifying that specific target state, the classical description complexity of the state itself, or the amount of information revealed when proving a property of that state (Buhrman et al., 2 Oct 2025, O'Donnell et al., 8 Jul 2025, Cai et al., 2015, Malavolta, 8 Oct 2025, Chia et al., 2024). This suggests that the term functions less as a single canonical invariant than as an umbrella for several related attempts to replace worst-case analysis by per-instance analysis in quantum settings.

1. Terminological scope and major interpretations

The literature currently exhibits several distinct but adjacent uses of the term. One line of work defines Quantum Instance Complexity as a description-length measure for a quantum solver on a fixed classical input $x$: the complexity is the size of the shortest classical program that generates a quantum circuit which is globally consistent with a language $L$ and actually decides $x$ (Buhrman et al., 2 Oct 2025). A second line uses the phrase in a resource-testing sense: for quantum state certification with entangled measurements, the optimal number of copies needed to certify a particular target state $\sigma$ depends on the spectral structure of that specific $\sigma$, not merely on ambient dimension (O'Donnell et al., 8 Jul 2025). A third line studies the complexity of describing or certifying individual quantum states themselves, most notably via tree size, a quantity attached to a single pure state $\ket{\psi}$ (Cai et al., 2015). Neighboring frameworks instead take the input instance to be a quantum state and study proof complexity, zero knowledge, or complexity-class structure without defining a single scalar “instance complexity” measure (Malavolta, 8 Oct 2025, Chia et al., 2024).

Usage	Instance object	Quantity studied
Descriptional quantum instance complexity	Classical string $L$0	Shortest classical description of a quantum $L$1-consistent solver
Instance-optimal certification	Target state $L$2	Copy complexity of certifying that specific $L$3
State description complexity	Pure or mixed quantum state	Tree size $L$4, $L$5, or related state-complexity notions
Quantum-state-input proof theory	Quantum state $L$6 or $L$7	Zero-knowledge, witness, interaction, and copy-access structure

The common theme is that complexity is attached to a particular object rather than to the worst case over all objects of a fixed size. What varies across the literature is which resource is being individualized: program length, sample complexity, description complexity of a state, or information leakage in verification.

2. Description-length formulations for fixed classical instances

The most explicit formalization appears in the framework of time-bounded classical and quantum instance complexity for a language $L$8 (Buhrman et al., 2 Oct 2025). The classical precursor is time-bounded instance complexity

$L$9

where $\sigma$0 is the shortest $\sigma$1-consistent program running in time $\sigma$2 such that $\sigma$3. Here $\sigma$4-consistency means that $\sigma$5 for all $\sigma$6, and whenever $\sigma$7, one has $\sigma$8. The crucial feature is the “I don’t know” output $\sigma$9: the program need not solve all inputs, only remain globally correct wherever it does answer.

The quantum version keeps the description itself classical. A program $\rho$0 outputs a classical description of a quantum circuit $\rho$1; the framework does not use quantum advice states in the definition. The paper first introduces time-bounded quantum Kolmogorov and distinguishing analogues, including

$\rho$2

for the shortest classical program that generates a circuit outputting $\rho$3 with probability $\rho$4, and

$\rho$5

for the shortest classical program generating a quantum circuit that accepts $\rho$6 with probability $\rho$7 and every $\rho$8 with probability $\rho$9. It then defines quantum $x$0-$x$1-consistency via two designated output qubits $x$2: $x$3 is a confidence flag, and whenever $x$4, the answer qubit must satisfy $x$5.

The central quantity is then

$x$6

the size of the shortest classical program $x$7 that runs in time $x$8, generates a quantum circuit, is quantum-$x$9-$L$0-consistent for some $L$1, and decides the chosen instance $L$2 (Buhrman et al., 2 Oct 2025). Because the bias parameter can be amplified by repetition and majority, the dependence on $L$3 is not essential up to polynomial overhead. The task is still decision, not witness generation or search.

Several comparison inequalities organize the framework. The paper states that

$L$4

It also emphasizes that for $L$5, there is a constant-size quantum description of the general polynomial-time quantum algorithm for all instances, so $L$6 is $L$7 for every $L$8. By contrast, if $L$9, there cannot be a uniform constant upper bound on classical $x$0 for all $x$1, since a finite list of constant-size classical programs could then be searched to obtain a polynomial-time classical decider for $x$2. Under $x$3, some instances must therefore have small quantum instance complexity and nonconstant classical instance complexity.

Within this formulation, “quantum instance complexity” is not a complexity of the input string alone. It is the complexity of the shortest useful instance-specific quantum algorithmic description for deciding membership of that string in a language.

3. Queasy instances, additive gaps, and reductions from factoring to SAT

The same framework introduces queasy instances as a formal notion of single-instance quantum advantage (Buhrman et al., 2 Oct 2025). An instance $x$4 is queasy with respect to $x$5 when the shortest efficient quantum description is significantly smaller than the shortest efficient classical $x$6-consistent description, with the time bounds taken asymmetrically: $x$7 The additive amount of queasiness is

$x$8

and the normalized queasiness factor is

$x$9

The paper further defines an instance to be maximally queasy when the upper bound

$\sigma$0

is saturated up to constants.

A key structural theorem states that low instance complexity implies wide algorithmic utility. If $\sigma$1 is a shortest $\sigma$2-consistent program deciding $\sigma$3, and

$\sigma$4

then for the set $\sigma$5 one has

$\sigma$6

for some constants $\sigma$7. The paper states that the same proof works quantumly with $\sigma$8 in place of $\sigma$9. This is the sense in which queasiness yields “exponential algorithmic utility”: a short instance-specific solver that is much shorter than a distinguisher must also answer on exponentially many other inputs.

The headline application uses a factoring-derived language

$\sigma$0

Because factoring is in BQP via Shor, $\sigma$1. Under the assumption that factoring on $\sigma$2-bit instances requires $\sigma$3 time for some $\sigma$4, the paper proves the existence of $\sigma$5 such that

$\sigma$6

for $\sigma$7 and $\sigma$8. It then transfers this gap to SAT via an invertible one-to-one polynomial-time reduction, yielding SAT formulas $\sigma$9 such that

$\ket{\psi}$0

for some $\ket{\psi}$1 and $\ket{\psi}$2.

This application does not place SAT in BQP. Rather, it identifies individual formulas whose structure is inherited from factoring through an invertible Cook–Levin-type reduction. Quantumly, these formulas are easy because a constant-size description can invoke the factoring routine; classically, short $\ket{\psi}$3-consistent deciders for too many such formulas would imply an unexpectedly strong classical factoring algorithm. The paper discusses these SAT instances as close to being maximally queasy, while the explicit theorem in the body provides a polynomial-versus-subexponential additive gap.

4. Instance-optimal quantum state certification and copy complexity

A different meaning of quantum instance complexity arises in quantum state certification with collective entangled measurements (O'Donnell et al., 8 Jul 2025). The task is: given an explicit description of a $\ket{\psi}$4-dimensional hypothesis state $\ket{\psi}$5 and $\ket{\psi}$6 identical copies of an unknown state $\ket{\psi}$7, decide with constant success probability between

$\ket{\psi}$8

The only counted resource is the number of copies $\ket{\psi}$9, and the tester may perform fully entangled measurements over all copies. Prior work established a worst-case copy complexity $L$00, with the lower bound coming from the maximally mixed instance $L$01. The paper resolves the open question of the correct instance-by-instance dependence when measurements are unrestricted.

The main result is a nearly instance-optimal upper/lower characterization in terms of the fidelity with the maximally mixed state, equivalently the Schatten-$L$02 quasinorm. In informal form, for $L$03,

$L$04

where $L$05 and $L$06 are trimmed-and-renormalized versions of $L$07. The lower bound removes $L$08 eigenvalue mass, while the upper bound removes $L$09 small mass. In the formal theorem statements, the lower bound uses a state $L$10 obtained by zeroing out at most $L$11 eigenvalue mass, and the upper bound uses a $L$12 obtained by zeroing out the lowest $L$13 total eigenvalue mass.

If $L$14, then

$L$15

so the governing parameter is

$L$16

Up to polylogarithmic factors and the trimming mismatch, the copy complexity is therefore

$L$17

This interpolation is sharp across basic spectral regimes. If $L$18 is pure, then $L$19, giving

$L$20

If $L$21, then $L$22, giving

$L$23

More generally, if $L$24 is close to maximally mixed on an $L$25-dimensional support with nonzero eigenvalues all $L$26, then $L$27, so

$L$28

Lower-rank or more peaked states are therefore easier to certify; states closer to maximally mixed are hardest.

The upper bound modifies the unentangled instance-optimal algorithm of Chen et al. by diagonalizing $L$29, bucketing eigenvalues into dyadic ranges

$L$30

removing a tail of total mass at most $L$31, and then replacing the previous unentangled Hilbert–Schmidt tester with the entangled-measurement tester $L$32, which distinguishes $L$33 from $L$34 using $L$35 copies. The lower bound introduces a quantum analogue of the Ingster–Suslina method based on the largest quantum $L$36-divergence

$L$37

together with the inequality

$L$38

For mixedness testing, Haar-random perturbations yield directly

$L$39

recovering the lower bound $L$40 by a proof the paper describes as surprisingly simple relative to earlier representation-theoretic arguments.

This is a strong form of quantum instance complexity because the optimal sample requirement is controlled by the fine spectral profile of the specific target state, not by dimension alone. The result is nevertheless only nearly instance-optimal: the bounds differ by polylogarithmic factors, and the lower and upper theorems use slightly different trimming procedures.

5. State-instance complexity via tree size

A third notion attaches complexity directly to an individual quantum state by asking how economically it can be written in a restricted classical syntax (Cai et al., 2015). For a pure $L$41-qubit state

$L$42

a tree representation is a rooted tree whose internal nodes are labeled by $L$43 or $L$44, and whose leaves are labeled by single-qubit states of the form $L$45. The tree size $L$46 is the minimum number of leaves over all such trees. The approximate version is

$L$47

and the paper also discusses a mixed-state extension

$L$48

Tree size is presented as a complexity measure for individual state instances. A family is simple if $L$49 grows polynomially in $L$50, and complex if $L$51 grows superpolynomially. Two properties are emphasized: tree size is computable in principle, and nontrivial lower bounds can be proved. The paper contrasts this with quantum Kolmogorov complexity, which is based on the shortest generating program and is not computable.

For small systems, the measure is exact and structurally informative. Two-qubit separable states have $L$52, entangled states have $L$53. For three qubits, the canonical representatives satisfy

$L$54

Thus the $L$55 class attains maximal exact three-qubit tree size, but $L$56 for arbitrarily small $L$57, because $L$58 can be approximated arbitrarily well by GHZ-class states. For four qubits, the maximal exact tree size is $L$59, while for $L$60 the maximal approximate tree size drops to $L$61.

The principal lower-bound mechanism is a reduction to multilinear formula size. To a state $L$62, the paper associates the function

$L$63

and proves

$L$64

Raz’s theorem on multilinear formulas then yields superpolynomial lower bounds from rank growth across random bipartitions. In state language, if the Schmidt rank across a random equal bipartition is sufficiently large with probability $L$65, then

$L$66

This machinery gives explicit superpolynomial lower bounds for several families. If all coefficients in an immanant are nonzero, then the corresponding immanant state has

$L$67

Subgroup states $L$68 have the same bound under an invertibility condition on random bipartition submatrices, and also satisfy an approximate lower bound

$L$69

Most prominently, the 2D cluster state has

$L$70

The operational relevance comes from measurement-based quantum computation. The paper proves that if an MBQC resource state has polynomial tree size, then the computation can be simulated efficiently classically. Superpolynomial tree size is therefore necessary for universal MBQC. At the same time, large tree size is not sufficient for computational advantage: subgroup states can have superpolynomial tree size while still being classically simulable by Gottesman–Knill methods. The paper also shows that large tree size can be certified efficiently for subgroup states and 2D cluster states via stabilizer witnesses such as

$L$71

6. Quantum-state-input complexity, zero knowledge, and neighboring frameworks

A neighboring body of work takes the instance itself to be a quantum state and studies class structure, proof systems, and knowledge leakage rather than a scalar instance-complexity measure (Malavolta, 8 Oct 2025, Chia et al., 2024). In this setting, a quantum decision problem is a promise problem

$L$72

whose instances are states $L$73 or $L$74. The verifier receives copies of the state itself, not a classical description. The classes $L$75, $L$76, $L$77, $L$78, $L$79, and $L$80 formalize the resulting complexity landscape for pure and mixed quantum inputs.

Several structural facts are relevant to the broader theme of instance-specific quantum complexity. One paper proves the containments

$L$81

but also establishes unconditional separations

$L$82

It further shows that interaction can reduce the effective difficulty of some state-property testing tasks by exponentially many samples relative to single-party models, using mixedness testing and maximally entangled state testing as examples (Chia et al., 2024). A natural implication is that access structure to the particular state instance—polynomial copies, unlimited copies, witnesses, or interaction—is itself part of the instance-specific complexity.

Another paper studies knowledge complexity for decision problems with quantum-state inputs. It emphasizes that this is not classical instance complexity in the Kolmogorov or partial-program sense. Instead, it asks how much additional information about a concrete quantum-state instance is revealed when one proves membership in a yes-set. Its main theorem shows that, assuming EFI pairs exist,

$L$83

so every pure-state-input quantum interactive proof can be transformed into a computational zero-knowledge proof with unconditional soundness against malicious verifiers. It also proves

$L$84

while showing a limitation for mixed-state instances: under LWE hardness, there exist problems in mQMA without an efficient zero-knowledge protocol (Malavolta, 8 Oct 2025). The same paper gives a protocol for the Uhlmann transformation problem in which the simulator uses exactly one oracle query to the Uhlmann unitary.

These frameworks clarify an important distinction. In description-length approaches, quantum instance complexity is the size of the shortest useful quantum solver description for a fixed classical input. In copy-complexity approaches, it is the optimal number of state copies needed for a particular target. In tree-size approaches, it is the classical expression complexity of a specific state. By contrast, in quantum-state-input proof theory, the central issue is often whether a property of a given state can be verified with limited copies, specific witness models, or zero additional knowledge leakage. The notions are adjacent, and they increasingly interact, but they are not interchangeable.

Across these usages, several unresolved themes recur: the dependence of complexity on copy access to the instance, the pure-versus-mixed distinction, the gap between exact and approximate notions, and the lack of a single universally adopted scalar measure covering all these settings. The present literature therefore supports a plural rather than singular understanding of quantum instance complexity.