Quantum Mean-Value Problem: Formulations & Methods

• The quantum mean-value problem encompasses various formulations for estimating expectation values, cost functions, and arithmetic means across different quantum models.
• It highlights the role of diverse access models and error criteria, with techniques ranging from additive error approximations to relative control methods.
• Multiple approaches, including classical algorithms, amplitude estimation, and qRAM-based methods, are employed to efficiently compute mean values in quantum systems.

The “Quantum Mean-Value Problem” does not denote a single universally agreed task. Taken together, the literature uses the label for several distinct formulations: the estimation or computation of an expectation value such as $\mu=\langle 0^n|U^\dagger O U|0^n\rangle$ for a quantum circuit output; the evaluation of a variational objective such as a QAOA cost expectation; the estimation of an arithmetic mean $\mu=\frac{1}{N}\sum_x f(x)$ or a vector mean $\vec\mu=\mathbb E[\vec X]$ by quantum query algorithms; and, in more mathematical settings, mean-value identities in optical tomography or $q$-calculus (Bravyi et al., 2019, Zhuang et al., 2021, Brassard et al., 2011, Cornelissen et al., 2021). The common thread is the extraction of an averaged quantity from quantum structure, but the access model, the error criterion, and even the meaning of “mean value” vary substantially across subfields.

1. Core formulations and problem classes

In quantum circuit complexity and variational algorithms, the central object is an expectation value of an observable in a prepared state. A canonical formulation is

$$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$

where $U$ is an $n$-qubit circuit and the $O_j$ are single-qubit operators (Bravyi et al., 2019). In QAOA, the same task appears as the cost expectation

$$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$

with $\hat C$ an Ising-type Hamiltonian and $\mu=\frac{1}{N}\sum_x f(x)$0 the depth-$\mu=\frac{1}{N}\sum_x f(x)$1 QAOA state (Zhuang et al., 2021).

In quantum query algorithms, the problem is instead to estimate a scalar or vector mean from oracle access. The scalar version takes a black-box function $\mu=\frac{1}{N}\sum_x f(x)$2 and targets

$\mu=\frac{1}{N}\sum_x f(x)$3

while multivariate versions target $\mu=\frac{1}{N}\sum_x f(x)$4 under coherent access to a random vector (Brassard et al., 2011, Cornelissen et al., 2021). In qRAM-based proposals, the mean is often encoded directly into an amplitude or measurement probability, with the data-loading primitive treated as a black box (Tamirat, 2022).

These formulations differ not only in their input models but also in their approximation notions. Some works seek additive error $\mu=\frac{1}{N}\sum_x f(x)$5, others relative control such as $\mu=\frac{1}{N}\sum_x f(x)$6, and others only the magnitude of a mean because the measurement probability gives $\mu=\frac{1}{N}\sum_x f(x)$7 rather than $\mu=\frac{1}{N}\sum_x f(x)$8 itself (Bravyi et al., 2019, Tamirat, 2022). The computational content of the “mean-value problem” is therefore inseparable from the specific oracle model and error metric.

2. Expectation values in shallow circuits, QAOA, and variational optimization

A major strand of the literature studies when quantum expectation values can be computed classically. For constant-depth circuits with tensor-product observables, one result gives a deterministic polynomial-time classical algorithm with small relative error when each $\mu=\frac{1}{N}\sum_x f(x)$9 is close to the identity, a $\vec\mu=\mathbb E[\vec X]$0-time additive approximation algorithm when the $\vec\mu=\mathbb E[\vec X]$1 are positive semidefinite, and a randomized linear-time additive approximation algorithm for geometrically local circuits on a two-dimensional grid with arbitrary bounded single-qubit observables (Bravyi et al., 2019). The algorithms respectively use Barvinok’s polynomial interpolation method, a polynomial approximation for the OR function arising from quantum query complexity, and a Monte Carlo method combined with Matrix Product State techniques (Bravyi et al., 2019).

For QAOA, the problem becomes more structured because the cost Hamiltonian decomposes into local Ising terms,

$\vec\mu=\mathbb E[\vec X]$2

so the total expectation decomposes into local correlators $\vec\mu=\mathbb E[\vec X]$3 and $\vec\mu=\mathbb E[\vec X]$4 (Zhuang et al., 2021). The graph-decomposition algorithm of (Zhuang et al., 2021) exploits the fact that, at shallow depth $\vec\mu=\mathbb E[\vec X]$5, only a radius-$\vec\mu=\mathbb E[\vec X]$6 neighborhood subgraph $\vec\mu=\mathbb E[\vec X]$7 or $\vec\mu=\mathbb E[\vec X]$8 can affect a given local observable. For bounded-degree or moderately dense graphs at fixed $\vec\mu=\mathbb E[\vec X]$9, each local term is computed on a constant-size reduced circuit and the overall runtime scales linearly with the number of qubits; the complete-graph case is the main excluded regime because the relevant neighborhoods become global (Zhuang et al., 2021). The implementation, QCover, was compared against IBM Qiskit state-vector simulation and Quimb tensor-network contraction on Max-Cut, graph coloring, and the Sherrington–Kirkpatrick model, with “orders of magnitude performance improvement” reported in the sparse cases (Zhuang et al., 2021).

This distinction between expectation-value evaluation and full-state simulation is central. The QAOA result does not imply that QAOA sampling is classically easy; it implies that, for shallow depth and sparse locality, the variational inner loop based on evaluating $q$0 is often classically tractable (Zhuang et al., 2021). The paper is explicit that “current NISQ processors has no advantages in quantum mean value problem in QAOA,” while leaving open the hardness of the final sampling task (Zhuang et al., 2021).

A related but different response to the expectation-value bottleneck appears in the “quantum mean value approximator” for fully connected integer-encoded Hamiltonians. There the exact QMV

$q$1

is structurally exponential because the relevant correlators involve sums over subsets of the remaining qubits, and locality-based classical methods do not apply (Joseph et al., 2021). The proposed surrogate objective

$q$2

keeps only the most significant qubits in the binary encoding, reducing asymptotic evaluation cost to $q$3 relative to the full evaluation while empirically preserving the minima relevant for parameter selection in shortest-vector-problem instances (Joseph et al., 2021).

Expectation values after short-time Hamiltonian evolution form another restricted but important regime. For bounded, geometrically local, time-dependent Hamiltonians on 2D or 3D lattices, a classical algorithm approximates

$q$4

for product initial states and product observables, provided the evolution time $q$5 is constant (Saem et al., 2023). The key ingredient is a Lieb–Robinson lightcone truncation

$q$6

which reduces the global problem to many local simulations; for polynomial-time evolution the paper says the problem is essentially BQP-complete, so the tractability claim is strictly a constant-time one (Saem et al., 2023).

3. Quantum algorithms for scalar arithmetic means

A second major strand asks whether a quantum computer can estimate an arithmetic mean faster than classical sampling. In the black-box model with $q$7, one algorithm prepares a superposition over inputs, encodes $q$8 into an ancilla amplitude, and applies amplitude estimation. The resulting estimator $q$9 outputs $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$0 such that

$$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$1

with probability at least $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$2, using $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$3 evaluations of $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$4; this matches the $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$5 lower bound of Nayak and Wu for additive error $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$6, so the algorithm is asymptotically optimal in query complexity (Brassard et al., 2011). The same paper also gives $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$7, a variation on an earlier algorithm due to Aharonov, which estimates the mean bit by bit using quantum counting and majority amplification; its error depends on the bit-profile of the inputs rather than a clean uniform additive parameter (Brassard et al., 2011).

A different line of work replaces amplitude estimation by qRAM-based interference. One proposal assumes access to amplitude-encoded value states

$$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$8

loads them in superposition, applies Hadamards to isolate the $$\mu=\langle 0^n|U^\dagger O U|0^n\rangle, \qquad O=O_1\otimes O_2\otimes\cdots\otimes O_n,$$9 branch, and arranges for the final output-qubit amplitude to equal

$U$0

The measurement probability is then $U$1, so repeated sampling estimates only $U$2 unless an additional sign-recovery procedure is supplied; the stated runtime is $U$3, but the result depends on a strong qRAM/state-preparation assumption and does not use amplitude estimation (Tamirat, 2022).

Another variant treats the mean of amplitudes of a normalized quantum state

$U$4

By engineering a start state whose marked branch amplitude is proportional to $U$5 and then applying a Grover-like amplification procedure, the method estimates the mean in $U$6 steps, compared with $U$7 classically for a structureless function, assuming efficient state preparation (Tucci, 2014). This formulation is distinct from oracle mean estimation because the input is a state-preparation circuit rather than query access to classical function values.

4. Multivariate, heavy-tailed, and non-identical mean estimation

Multivariate mean estimation is more subtle than the univariate case because a coordinatewise reduction does not preserve the quantum advantage. One paper develops the first near-optimal quantum algorithms for estimating $U$8 in Euclidean norm in two input models, a binary-oracle model and a weaker phase-oracle model (Cornelissen et al., 2021). In the binary-oracle model, the main high-precision result gives

$U$9

hence

$n$0

for $n$1, while matching lower bounds show that no quantum improvement over the classical $n$2 rate is possible when $n$3 (Cornelissen et al., 2021). The same paper proves that the phase-oracle model is strictly weaker for mean estimation (Cornelissen et al., 2021).

A 2025 refinement improves the multivariate complexity by polylogarithmic factors using a generalized Grover operator as a mean-value phase oracle. The main theorem gives an estimator $n$4 with

$n$5

and hence

$n$6

with probability at least $n$7, using

$n$8

accesses to the quantum experiment; a lower-memory variant uses $n$9 quantum registers but costs an extra $O_j$0 (Tang, 9 Apr 2025). The remaining $O_j$1 factor is attributed to the phase-estimation primitive (Tang, 9 Apr 2025).

Heavy-tailed scalar mean estimation also admits a quantum analogue of robust sub-Gaussian estimation. A quantum estimator for a q-random variable with mean $O_j$2 and variance $O_j$3 satisfies

$O_j$4

using

$O_j$5

quantum experiments, without prior variance information and under only finite-variance assumptions (Hamoudi, 2021). The same work proves an $O_j$6 lower bound, so the complexity is optimal up to polylogarithmic factors (Hamoudi, 2021).

The non-identical setting changes the picture again. When the algorithm receives access to a sequence of different random variables $O_j$7 with means all $O_j$8-close to a common target $O_j$9, bounded and sub-Gaussian families still admit quantum estimators with essentially quadratic improvement over classical sample complexity, but a generic quadratic speedup is impossible in the absence of such structure (Hu et al., 2024). The technical reason is that direct amplitude estimation does not work with non-identical query access; the paper overcomes this in the upper bounds by reducing bounded and sub-Gaussian variables to the Bernoulli case and using an uncomputation trick, while the lower bounds are proved by simulating non-identical oracles by parallel oracles and by an adversarial method with non-identical oracles (Hu et al., 2024).

Earlier qRAM-based multivariate proposals already aimed to estimate a mean vector $$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$0 by interference and entanglement, with a stated complexity $$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$1. Their mechanism makes the componentwise means appear as amplitudes and then as probabilities $$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$2, but the analysis is informal, relies on normalized amplitude-encoded vectors, and does not resolve signs directly (Tamirat, 2019).

5. Mean values in semiclassical dynamics, lattice field theory, and quantum observables

In semiclassical chaos, the quantum mean-value problem becomes the computation of

$$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$3

for a smooth detector observable in a regime where $$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$4 and the classical dynamics is strongly chaotic. One approach rewrites the exact mean in Wigner–Weyl form and approximates it by a semiclassical mean-value formula

$$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$5

where $$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$6 labels interfering filament pairs of the evolved Lagrangian manifold (Lando et al., 2023). The paper shows that in the deep chaotic regime this direct mean-value representation remains accurate while the Herman–Kluk propagator produces essentially numerical noise (Lando et al., 2023).

In Euclidean lattice field theory, observables of the form

$$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$7

are recast as quantum mean-estimation tasks by encoding $$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$8 as a phase and then applying a QPE-based quantum mean estimation algorithm (Gustafson et al., 2023). The paper argues that the resulting complexity gives a quadratic advantage over Monte Carlo, that the advantage persists even in the presence of a sign problem, and that the method is structurally insensitive to critical slowing down because it avoids Markov-chain sampling (Gustafson et al., 2023). The practical bottleneck is coherent state preparation and arithmetic-heavy phase oracles, not the mean-estimation principle itself (Gustafson et al., 2023).

A more operator-theoretic example is the mean value of the quantum potential. For a pure state $$E_p(\vec\gamma,\vec\beta)=\langle \vec\gamma,\vec\beta|\hat C|\vec\gamma,\vec\beta\rangle,$$9 and

$\hat C$0

the paper proves the general lower bound

$\hat C$1

for any $\hat C$2, and identifies the mean quantum potential with the nonclassical part of momentum correlations,

$\hat C$3

(2002.01507). In one dimension this yields a generalized uncertainty relation stronger than Robertson–Schrödinger in the sense described in the paper (2002.01507).

6. Other meanings: optical tomography, $\hat C$4-calculus, and adjacent “mean” problems

A distinct mathematical use of “quantum mean value” appears in optical tomography. Given an optical tomogram

$\hat C$5

a dual map sends observables $\hat C$6 to generalized functions

$\hat C$7

so that

$\hat C$8

for a large class of observables including symmetrized polynomials in $\hat C$9 and $\mu=\frac{1}{N}\sum_x f(x)$00 (Amosov et al., 2011). Here the “mean-value problem” is the direct recovery of observable expectations from tomographic probability distributions.

In Jackson $\mu=\frac{1}{N}\sum_x f(x)$01-calculus, the term refers to a $\mu=\frac{1}{N}\sum_x f(x)$02-analogue of Lagrange’s mean value theorem rather than quantum-state expectation values. The paper proves that the naive classical mean value theorem is false in shifted Jackson $\mu=\frac{1}{N}\sum_x f(x)$03-calculus, but that a correct statement holds along a $\mu=\frac{1}{N}\sum_x f(x)$04-chain: $\mu=\frac{1}{N}\sum_x f(x)$05 for some $\mu=\frac{1}{N}\sum_x f(x)$06 when $\mu=\frac{1}{N}\sum_x f(x)$07 is continuous (Aljinović et al., 2020). That corrected theorem then yields sharp $\mu=\frac{1}{N}\sum_x f(x)$08-Ostrowski and midpoint inequalities (Aljinović et al., 2020).

A related but distinct literature concerns the quantum mean-field problem rather than expectation-value estimation. One example introduces a quantum Wasserstein-type functional $\mu=\frac{1}{N}\sum_x f(x)$09 and proves a Dobrushin-type estimate comparing the $\mu=\frac{1}{N}\sum_x f(x)$10-body Schrödinger dynamics with Hartree dynamics, with error controlled uniformly in the classical limit $\mu=\frac{1}{N}\sum_x f(x)$11 for $\mu=\frac{1}{N}\sum_x f(x)$12 interaction potentials (Golse et al., 2015). This is not a mean-value algorithmic problem, but it shows that “mean” in quantum theory also names asymptotic collective descriptions rather than observable averages.

Taken together, these lines of work show that the quantum mean-value problem is best understood as a family of problems organized by access model and physical setting. In shallow-circuit simulation the decisive issues are locality, depth, and observable structure; in quantum query algorithms they are oracle power, tail assumptions, and dimension; in semiclassical and field-theoretic physics they are phase-space geometry, interference structure, and state preparation; and in mathematical formulations such as optical tomography or $\mu=\frac{1}{N}\sum_x f(x)$13-calculus the term denotes exact integral or differential identities rather than computational speedups.