Expressibility, Entangling Capability, and Trainability Trade-offs

A parameterized quantum circuit $U(\boldsymbol\theta)$ is not one unitary but a family of them, and every family carries three numbers on its back. Expressibility asks how thoroughly the output states $|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle$ cover Hilbert space as the angles are swept — do they blanket it like the uniform Haar distribution, or huddle in a patch? Entangling capability asks how much multipartite correlation the family typically produces. Trainability asks the practical question that decides whether the circuit is any use: are the cost gradients large enough to optimize with a realistic number of measurements? The surprise, and the whole subject of this article, is that these are not three free dials. The very property that certifies maximal expressibility — the output ensemble reproducing the second moment of the Haar measure, i.e. forming an approximate unitary $2$ -design — is word for word the hypothesis of the barren-plateau theorem, under which the expected gradient vanishes and its variance shrinks like $2^{-n}$ in the qubit number $n$ . Entanglement is the bridge: Haar-typical states carry near-maximal average entanglement, and the directions that generate it are precisely the ones whose gradients disappear. The design lesson is sharp and counterintuitive — choose the least expressible ansatz that can still reach the target — reframing ansatz engineering as navigation of a Pareto frontier, not a race toward universality.

Key takeaways

Expressibility measures how uniformly an ansatz's output states cover Hilbert space — the deviation of its second frame moment (or its fidelity distribution) from the Haar value; entangling capability is the ensemble-averaged Meyer–Wallach entanglement.
Both come from the same $t=2$ statistic of the circuit ensemble.
High expressibility is mathematically tied to barren plateaus — an approximate unitary $2$ -design has cost-gradient variance $\mathcal{O}(2^{-n})$ , an exponentially flat landscape.
The trade-off is therefore structural: an ansatz expressible enough to reach any target is generically untrainable, while a restricted, problem-tailored ansatz keeps gradients alive by sacrificing coverage.
Good design navigates the frontier with shallow depth, locality, symmetry — and local cost observables, since a global cost plateaus even at constant depth.
Key idea: the property that makes an ansatz a universal approximator — its output ensemble approximating a unitary $2$ -design — is identical to the hypothesis that flattens its landscape, so expressibility and trainability trade off through the single shared quantity of the second frame moment.

The core insight

The property that makes an ansatz a universal approximator — its output ensemble approximating a unitary $2$ -design — is identical to the hypothesis that flattens its optimization landscape; expressibility and trainability therefore trade off through the shared quantity of the second frame moment $\|\mathcal{A}^{(2)}\|$ .

Where this lives today

The trade-off is the daily reality of variational quantum algorithms. In quantum chemistry, hardware-efficient ansätze were the first instinct — cheap, universal — and the first casualties, since their expressibility grows with depth straight into a plateau; the field has swung to chemistry-inspired ansätze (UCCSD, ADAPT-VQE) that restrict expressibility with particle-number and spin symmetry, buying $\mathrm{poly}(n)$ gradients at the cost of universality. QAOA occupies a favorable niche for the same reason: its alternating problem/mixer layers with shared angles form a constrained, low-expressibility manifold whose $\|\mathcal{A}^{(2)}\|$ stays far from zero, which is why it often trains where a generic ansatz is already flat. And the mitigation frontier — small-angle initialization, layerwise and local-cost training, symmetry-equivariant circuits — is, at heart, a toolkit for holding a circuit at a chosen point on that Pareto frontier rather than sliding off its edge.

The physical picture

Imagine the reachable output states of an ansatz as a scatter of points on the unit sphere of Hilbert space. A shallow, restricted circuit paints a small patch — it reaches only states near where it started, but its cost surface still has honest hills and valleys. Deepen and entangle the circuit and the scatter spreads until the points sprinkle everywhere with the density of the uniform Haar measure. Universality wants that full coverage — but a cloud that looks the same in every direction makes the cost look the same in every direction too, so on average its slope is zero and its fluctuations shrink with the space's dimension $2^{n}$ . Coverage and slope pull against each other.

Figure

Two ansätze on the Hilbert-space sphere. Left, a restricted circuit covers a small patch — limited reach but honest gradients, Var ~ 1/poly(n). Right, a deep 2-design sprinkles states uniformly with Haar mean overlap ⟨F⟩ = 1/2ⁿ — full coverage but an exponentially flat landscape, Var ~ 2⁻ⁿ

The bridge between coverage and slope is a single scalar. Read the "clumpiness" of the state cloud through its frame potential: the more Haar-like the cloud, the smaller the second frame potential, whose unique minimizer is the Haar value $\mathcal{F}^{(2)}=2$ . Barren plateaus switch on exactly when the clumpiness bottoms out there, because the ensemble's second moment is what controls a gradient's variance. So one number — the norm $\|\mathcal{A}^{(2)}\|$ of the deviation from Haar — simultaneously certifies expressibility and predicts flatness, which is why the trade-off is an identity, not a coincidence. The hardware consequence is brutal: since the typical gradient falls as $2^{-n/2}$ , the shots to resolve it climb as $2^{n}$ — the wall the second figure plots.

Figure

The barren-plateau cliff. For a 2-design ansatz the cost-gradient variance decays as Var ≈ 2⁻ⁿ, a straight line on a log axis: 9.8×10⁻⁴ at n = 10 falls to 9.5×10⁻⁷ at n = 20. Resolving that gradient needs ~2ⁿ shots — feasible at n = 10, but ~10¹⁵ shots (millennia of runtime) by n = 50

Entanglement drives the cloud from patch to sprinkle. A product-state ansatz has zero entanglement, is classically simulable, and is trivially trainable — and useless. A scrambling ansatz reaches volume-law entanglement and inherits both the design property and its plateau. Workable ansätze live in the narrow band between: enough entangling capability to reach the physics of interest, not so much that the ensemble turns Haar-typical.

The core equations

Expressibility is defined by comparing the ansatz's $t$ -th moment with the Haar one, packaged in a single frame-deviation operator that is zero exactly when they match. The order $t=2$ matters because it is at once the smallest moment that sees pairwise fidelities and the moment that governs gradient variances.

\mathcal{A}^{(t)}=\int_{\mathrm{Haar}}\!\big(|\psi\rangle\langle\psi|\big)^{\otimes t}\,\mathrm{d}\psi\;-\;\int_{\Theta}\!\big(|\psi_{\boldsymbol\theta}\rangle\langle\psi_{\boldsymbol\theta}|\big)^{\otimes t}\,\mathrm{d}\boldsymbol\theta \qquad (1)

where: $\mathcal{A}^{(t)}$ — $t$ -th frame-deviation operator; $\|\mathcal{A}^{(t)}\|$ is the expressibility measure ( $0$ = perfectly Haar at order $t$ ); $|\psi_{\boldsymbol\theta}\rangle=U(\boldsymbol\theta)|\psi_0\rangle$ — output state; $\mathrm{d}\psi$ — Haar measure on pure states; $\mathrm{d}\boldsymbol\theta$ — sampling measure over $\Theta$ . Smaller norm means a more Haar-like cloud, hence more expressible. In practice one samples pairs of parameter vectors and histograms their fidelity rather than diagonalizing $\mathcal{A}^{(2)}$ . For two independent Haar-random states the fidelity obeys a purely dimension-dependent law — the benchmark every ansatz is judged against.

P_{\mathrm{Haar}}(F)=(N-1)(1-F)^{\,N-2},\qquad N=2^{n},\qquad \langle F\rangle_{\mathrm{Haar}}=\frac{1}{N} \qquad (2)

where: $F=|\langle\psi|\phi\rangle|^{2}$ — fidelity of two sampled states; $N=2^{n}$ — Hilbert-space dimension; $P_{\mathrm{Haar}}$ — its density, piled up near $F=0$ with mean $2^{-n}$ : Haar states are nearly orthogonal. The expressibility of a concrete ansatz is then the statistical distance between its empirical fidelity histogram and $P_{\mathrm{Haar}}$ . Sim, Johnson and Aspuru-Guzik use the Kullback–Leibler divergence, in which smaller means more expressible.

\mathrm{Expr}=D_{\mathrm{KL}}\!\big(P_{\mathrm{ansatz}}(F)\;\|\;P_{\mathrm{Haar}}(F)\big) \qquad (3)

where: $\mathrm{Expr}$ — expressibility ( $\to 0$ is maximal); $P_{\mathrm{ansatz}}$ — empirical fidelity distribution; $D_{\mathrm{KL}}$ — KL divergence. An idle circuit returning $|\psi_0\rangle$ has $P_{\mathrm{ansatz}}(F)=\delta(F-1)$ and a huge divergence; a Haar-like circuit drives $\mathrm{Expr}\to 0$ . Entangling capability is tracked by the Meyer–Wallach measure, which averages the single-qubit "how mixed am I" over all qubits. An unentangled qubit is pure and contributes nothing; a maximally entangled one is maximally mixed and contributes the most.

Q(|\psi\rangle)=\frac{2}{n}\sum_{k=1}^{n}\big(1-\mathrm{Tr}\,\rho_k^{2}\big) \qquad (4)

where: $Q\in[0,1]$ — global entanglement, $0$ for product states; $\rho_k$ — reduced single-qubit state of qubit $k$ ; $\mathrm{Tr}\,\rho_k^{2}$ — its purity ( $=1$ iff unentangled, $=\tfrac12$ iff maximally mixed); $n$ — qubit count. The entangling capability of an ansatz is the ensemble average of $Q$ over the sampling distribution — one scalar locating the circuit on the entanglement axis.

\mathrm{Ent}=\frac{1}{|S|}\sum_{\boldsymbol\theta\in S}Q\big(|\psi_{\boldsymbol\theta}\rangle\big) \qquad (5)

where: $\mathrm{Ent}$ — averaged entangling capability; $S$ — sampled parameter set of size $|S|$ ; $Q(|\psi_{\boldsymbol\theta}\rangle)$ — the measure of each output. Empirically $\mathrm{Ent}$ rises with depth toward the Haar value. The trainability side is the barren-plateau statement. When the ansatz forms an $\varepsilon$ -approximate unitary $2$ -design, the cost gradient has zero mean and a variance capped by an exponentially small number.

\langle\partial_\mu C\rangle_{\boldsymbol\theta}=0,\qquad \mathrm{Var}_{\boldsymbol\theta}\!\left[\partial_\mu C\right]\;\le\;\frac{c(n)}{2^{\,n}} \qquad (6)

where: $C=\langle\psi_0|U^\dagger O\,U|\psi_0\rangle$ — cost (expectation of a Hermitian $O$ ); $\partial_\mu C$ — derivative in $\theta_\mu$ ; $c(n)$ — an $O(\mathrm{poly}\,n)$ prefactor. The landscape is flat in the mean and exponentially flat in its fluctuations. The unifying object is the frame potential — the ensemble-averaged squared overlap of two sampled unitaries. The Haar measure minimizes it at $t!$ , and equality defines an exact $t$ -design.

\mathcal{F}^{(t)}_{\mathcal{E}}=\int_{\mathcal{E}}\!\int_{\mathcal{E}}\big|\mathrm{Tr}(U^\dagger V)\big|^{2t}\,\mathrm{d}U\,\mathrm{d}V\;\ge\;\mathcal{F}^{(t)}_{\mathrm{Haar}}=t!\,,\qquad \mathcal{F}^{(2)}_{\mathrm{Haar}}=2 \qquad (7)

where: $\mathcal{F}^{(t)}_{\mathcal{E}}$ — $t$ -th frame potential; $t!$ — its floor, saturated iff $\mathcal{E}$ is a $t$ -design; $\mathrm{Tr}(U^\dagger V)$ — overlap of two sampled unitaries. For $t=2$ the floor is $2$ ; touching it is the barren-plateau regime. Assembled, the measures collapse onto one chain — the trade-off in a single line.

\mathrm{Expr}\downarrow\;\Longleftrightarrow\;\big\|\mathcal{A}^{(2)}\big\|\downarrow\;\Longleftrightarrow\;\mathcal{F}^{(2)}_{\mathcal{E}}\to 2\;\Longrightarrow\;\mathrm{Var}[\partial_\mu C]\to 2^{-n} \qquad (8)

where: $\downarrow$ — decreasing toward the Haar limit; expressibility cannot reach its ceiling without dragging the gradient variance to its floor.

Derivation AWhy the mean gradient vanishes for a one-design

The goal is to prove the cleaner half of the barren-plateau result: whenever a block of the circuit forms a unitary $1$ -design, the expected cost gradient is exactly zero — not merely small — so the landscape has no average descent direction and every scrap of optimization signal must live in the variance, using nothing beyond the first-moment integral over the unitary group.

Step 1. Write $C=\langle\psi_0|U^\dagger O\,U|\psi_0\rangle$ and split the circuit around the parameter of interest, $U=U_R\,V_\mu\,U_L$ with $V_\mu=e^{-i\theta_\mu H_\mu/2}$ and $H_\mu$ a Hermitian (Pauli) generator. Differentiating brings down a commutator, $\partial_\mu C=\tfrac{i}{2}\langle\psi_0|U_L^\dagger[H_\mu,\,U_R^\dagger O\,U_R]U_L|\psi_0\rangle$ . Why: the derivative of a conjugation is always a commutator with the generator — this is where the generator's algebra enters.

Step 2. Average over the ensemble, taking the right block $U_R$ to be a $1$ -design. The only $U_R$ -dependent factor, $U_R^\dagger O\,U_R$ , then averages to a multiple of the identity. Why: a $1$ -design is by definition an ensemble under which this integral forgets everything about $O$ except its trace.

\int U_R^\dagger\,O\,U_R\,\mathrm{d}\mu(U_R)=\frac{\mathrm{Tr}\,O}{N}\,\mathbb{1}

Step 3. Substitute back. The averaged operator is proportional to the identity, and the identity commutes with every generator, so $[\,\tfrac{\mathrm{Tr}\,O}{N}\mathbb{1},\,H_\mu]=0$ and $\langle\partial_\mu C\rangle=0$ exactly. Why: the mean gradient rested on $O$ keeping some directional structure after averaging; the $1$ -design erases it.

Step 4. A symmetric-in-the-mean landscape is not yet a plateau — a function can average to zero slope while keeping large local slopes. What kills training is that the variance also collapses: promoting the assumption to a $2$ -design and repeating at second order (a Weingarten pairing needing two copies of $U$ ) yields $\mathrm{Var}[\partial_\mu C]=\mathcal{O}(2^{-n})$ . Why: the mean uses one copy of $U$ , the variance uses two — exactly why $t=2$ is where expressibility and trainability meet.

Go deeper · Derivation B · 1 minHow expressibility upper-bounds the gradient variance

Derivation BHow expressibility upper-bounds the gradient variance

The goal is to upgrade the trade-off from a slogan to an inequality, so that measuring $\|\mathcal{A}^{(2)}\|$ on a few qubits predicts the gradient magnitude. Following Holmes, Sharma, Cerezo and Coles, the variance splits into a tiny Haar floor plus a term proportional to the frame deviation — so a circuit has appreciable gradients only if it is appreciably inexpressible.

Step 1. Write the variance as a genuine second moment, $\mathrm{Var}[\partial_\mu C]=\mathbb{E}_{\boldsymbol\theta}[(\partial_\mu C)^2]-(\mathbb{E}_{\boldsymbol\theta}[\partial_\mu C])^2$ . Both terms are quadratic in the output projector, i.e. linear functionals of the $t=2$ moment. Why: this is the formal reason the whole analysis is a statement about the second moment and nothing higher.

Step 2. Insert the frame-deviation operator (1) at $t=2$ : the ansatz second moment equals the Haar second moment minus the deviation, so any quadratic functional splits into a Haar piece and an $\mathcal{A}^{(2)}$ piece. Why: $\mathcal{A}^{(2)}$ is defined precisely so that "ansatz = Haar − deviation" holds operator-wise, making the split exact.

Step 3. Bound the pieces. The Haar piece is the McClean floor, $O(\mathrm{poly}\,n)/2^{n}$ , exponentially small on its own; the deviation piece is bounded by $\|\mathcal{A}^{(2)}\|$ times an observable-dependent factor $g(n)$ . Why: Cauchy–Schwarz against the fixed cost operator turns "how far from Haar" into a multiplicative bound on the surviving variance.

\mathrm{Var}_{\boldsymbol\theta}\!\left[\partial_\mu C\right]\;\le\;g(n)\,\big\|\mathcal{A}^{(2)}\big\|\;+\;\frac{c(n)}{2^{\,n}}

Step 4. Read the trade-off off the inequality. As the ansatz becomes more expressible, $\|\mathcal{A}^{(2)}\|\to 0$ and the bound collapses onto the bare $2^{-n}$ floor — high expressibility forces small gradients. Appreciable gradients are permitted only when $\|\mathcal{A}^{(2)}\|$ is large, i.e. when the circuit is deliberately kept far from a $2$ -design. Why: trainability is a budget — you spend inexpressibility to buy gradient magnitude, and a $2$ -design has spent it all.

Limiting cases

How the trade-off behaves in the regimes an ansatz designer meets:

Case	Result	Meaning
Shallow / local, local cost ( $L=\mathcal{O}(1)$ )	$\mathrm{Var}[\partial_\mu C]=\Omega(1/\mathrm{poly}(n))$	Low expressibility, restricted light-cones; gradients decay only polynomially — trainable, but covers a limited submanifold.
Deep / global ( $2$ -design)	$\mathrm{Var}[\partial_\mu C]\sim 2^{-n}$	Maximal pairwise expressibility ( $\mathcal{F}^{(2)}\to 2$ ) forces an exponentially flat landscape — a plateau; reachability is high but optimization fails.
Global cost observable	$\mathrm{Var}[\partial_\mu C]\sim 2^{-n}$ even at $L=\mathcal{O}(1)$	A cost coupling all $n$ qubits plateaus even a shallow circuit: trainability depends on the observable's locality, not only the ansatz.
Product-state ansatz ( $\mathrm{Ent}=0$ )	large gradients, zero reach	Trivially trainable and classically simulable, but cannot represent correlated targets — useless at the other extreme.
Clifford / free-fermion ansatz	$\mathrm{poly}(n)$ gradients	A restricted (non-universal) gate set keeps $\mathcal{F}^{(2)}$ far above the floor and the circuit simulable — trainable by construction.

Common trapthe misconception that "more expressible" is always better

It is natural to think a more expressible ansatz is strictly better — it reaches more states, so surely it can only help. The trap: the same knob you turn to gain reach (more depth, more entangling layers) drives the ensemble toward a unitary $2$ -design, and a $2$ -design is exactly the barren-plateau hypothesis. The reachability you gained is worthless if the landscape leading to it is exponentially flat — you can represent the answer but never find it. Expressibility is a resource to be spent sparingly, not maximized: choose the least expressible ansatz that can still represent your target, and reach for locality, symmetry, and shallow depth precisely because they hold $\|\mathcal{A}^{(2)}\|$ away from zero.

Worked examples

1 · Entangling capability from the Meyer–Wallach measure (fully worked)

Compute $Q$ for a product state, a two-qubit Bell state, an $n$ -qubit GHZ state, and the three-qubit W state.

Step 1. Product state. Every qubit is pure, so each $\rho_k$ has purity $\mathrm{Tr}\,\rho_k^2=1$ and each term $1-\mathrm{Tr}\,\rho_k^2=0$ . Thus $Q=\mathbf{0}$ — the trivial, trainable, useless corner.

Step 2. Bell state $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$ . Tracing out either qubit gives the maximally mixed $\rho_k=\mathbb{1}/2$ , purity $\tfrac12$ , so each term is $\tfrac12$ . With $n=2$ : $Q=\tfrac{2}{2}(\tfrac12+\tfrac12)=\mathbf{1}$ — maximally entangled by this measure.

Step 3. GHZ state $(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})/\sqrt2$ . Each single-qubit reduced state is again $\mathbb{1}/2$ , purity $\tfrac12$ , so $Q=\tfrac{2}{n}\cdot n\cdot\tfrac12=\mathbf{1}$ for every $n$ — the maximum, despite GHZ being famously fragile, since $Q$ measures single-qubit mixedness, not robustness.

Step 4. W state $(|100\rangle+|010\rangle+|001\rangle)/\sqrt3$ . Each qubit is $|1\rangle$ with probability $\tfrac13$ , giving $\rho_k=\tfrac23|0\rangle\langle0|+\tfrac13|1\rangle\langle1|$ and purity $(\tfrac23)^2+(\tfrac13)^2=\tfrac59$ . Each term is $\tfrac49$ , so $Q=\tfrac{2}{3}\cdot3\cdot\tfrac49=\tfrac{8}{9}\approx\mathbf{0.889}$ — highly but not maximally entangled. Takeaway: $Q=0$ (product) $<0.889$ (W) $<1$ (Bell, GHZ); an ansatz's $\mathrm{Ent}$ is the sample-average of exactly this number, and it climbs with depth toward the Haar value as the circuit nears a $2$ -design.

2 · The shot budget of a barren plateau (fully worked)

A hardware-efficient ansatz has reached the $2$ -design regime, so $\mathrm{Var}[\partial_\mu C]\approx 2^{-n}$ and $\langle\partial_\mu C\rangle=0$ . How many shots resolve a gradient against shot noise at $n=4,10,20,50$ ?

Step 1. A typical gradient is the standard deviation, $|\partial_\mu C|\sim\sqrt{\mathrm{Var}}=2^{-n/2}$ . Estimating an expectation to precision $\epsilon$ costs $M\sim 1/\epsilon^2$ shots, so seeing the gradient needs $\epsilon\lesssim 2^{-n/2}$ , i.e. $M\sim 2^{n}$ shots per component.

Step 2. Evaluate the wall. $n=4$ : $M\sim\mathbf{16}$ shots — trivial. $n=10$ : $M\sim\mathbf{1.0\times10^{3}}$ — routine. $n=20$ : $M\sim\mathbf{1.0\times10^{6}}$ — a million shots per component, per step. $n=50$ : $M\sim\mathbf{1.1\times10^{15}}$ shots.

Step 3. At an optimistic $10^{4}$ executions per second, the $n=50$ figure takes $\approx1.1\times10^{11}$ s $\approx$ 3600 years — for one gradient component at one step, before averaging over the hundreds of parameters a real optimization needs.

Step 4. The plateau is not a slow-convergence nuisance a better optimizer fixes; it is an exponential measurement wall baked into the ansatz's expressibility. Because it is set by $\|\mathcal{A}^{(2)}\|\to0$ , the only cure is to keep the circuit out of the $2$ -design regime — shallow depth, local cost, symmetry, or a warm start. Takeaway: $M\sim2^{n}$ makes generic deep hardware-efficient ansätze untrainable by $n\approx40$ – $50$ , regardless of optimizer quality.

Physics of the portrait

The trade-off has no single discoverer — it emerged when three separate lines of work (unitary designs, entanglement measures, and variational-algorithm practice) were seen to describe one object. Its dramatic centre is the barren plateau: the 2018 result that random deep circuits are exponentially flat.

Origins · 1 minbarren plateau and how this was discovered

barren plateau

the exponentially flat landscapes of 2-design ansätze, named in 2018 · variational quantum algorithms and circuit design

The barren plateau is the name Jarrod McClean, Sergio Boixo, Vadim Smelyanskiy, Ryan Babbush and Hartmut Neven gave, in 2018, to a phenomenon that quietly threatened the near-term quantum-computing program. They proved that if a randomly initialized parameterized circuit is expressive enough to form an approximate unitary $2$ -design, the expectation of every cost gradient is zero and its variance decays as $2^{-n}$ — the landscape flattens exponentially as qubits are added, so gradient-based training stalls before it starts. The result rests on unitary $t$ -designs, formalized in the 2000s as finite ensembles reproducing the low moments of the Haar measure, and on the frame potential that grades an ensemble's closeness to Haar; the entangling-capability axis was supplied by David Meyer and Nolan Wallach's 2002 single-scalar measure $Q$ . Sukin Sim, Peter Johnson and Alán Aspuru-Guzik made expressibility and entangling capability catalogue-able in 2019; Marco Cerezo and collaborators tied plateaus to cost locality in 2021; and Zoë Holmes, Kunal Sharma, Cerezo and Patrick Coles closed the loop in 2022, bounding the gradient variance directly by the expressibility. What began as three disconnected ideas is now one inequality: reach, entanglement, and trainability read off the same second moment.

History

2002

— Meyer and Wallach define a single-scalar global multipartite entanglement measure Q

2018

— McClean, Boixo, Smelyanskiy, Babbush and Neven name the barren plateau for 2-design circuits

2019

— Sim, Johnson and Aspuru-Guzik make expressibility and entangling capability quantitative

2021

— Cerezo and co-workers link plateaus to cost-function locality, not just depth

2022

— Holmes, Sharma, Cerezo and Coles bound gradient variance directly by expressibility

Going further

Texts and reviews. Cerezo et al., Variational Quantum Algorithms, Nat. Rev. Phys. 3, 625 (2021) — the standard survey · Preskill, Quantum Information lecture notes (Caltech Ph219) — Haar measure, unitary designs, frame potentials · Watrous, The Theory of Quantum Information (2018), Ch. 7 — moment operators and the unitary group · Nielsen and Chuang, Quantum Computation and Quantum Information, Ch. 2, 8 — density matrices, purity, reduced states underlying $Q$ .

Primary sources. Sim, Johnson and Aspuru-Guzik, Adv. Quantum Technol. 2, 1900070 (2019) — expressibility and entangling-capability measures · McClean, Boixo, Smelyanskiy, Babbush and Neven, Nat. Commun. 9, 4812 (2018) — barren plateaus and $2$ -designs · Holmes, Sharma, Cerezo and Coles, PRX Quantum 3, 010313 (2022) — expressibility bounds on gradients · Cerezo, Sone, Volkoff, Cincio and Coles, Nat. Commun. 12, 1791 (2021) — cost-function-dependent barren plateaus · Meyer and Wallach, J. Math. Phys. 43, 4273 (2002) — the measure $Q$ . The key results collected in this article are: the Haar fidelity density $P(F)=(N-1)(1-F)^{N-2}$ with mean $1/N$ , $N=2^n$ ; the frame floor $\mathcal{F}^{(t)}_{\mathrm{Haar}}=t!$ and its saturation defining $t$ -designs; the vanishing mean and $\mathcal{O}(2^{-n})$ variance for $2$ -designs; the expressibility–variance inequality $\mathrm{Var}\le g(n)\|\mathcal{A}^{(2)}\|+c(n)2^{-n}$ ; the Meyer–Wallach values $Q=0$ (product), $8/9$ (W₃), $1$ (Bell, GHZ); and the shot scaling $M\sim2^{n}$ from $\sqrt{\mathrm{Var}}\sim2^{-n/2}$ . All scalings are asymptotic orders; no fabricated constants.

Prerequisites. Parameterized Quantum Circuits and Ansatz Design · Barren Plateaus in Variational Landscapes · Cost-Function and Measurement Estimation · Parameter-Shift Rules and Quantum Gradients. Leads to. Strategies to Mitigate Barren Plateaus · Hardware-Efficient and Chemistry-Inspired Ansätze · Quantum Machine Learning Models · QAOA Performance Bounds and Concentration.

Ask the tutor: Ask me to walk through why deepening a hardware-efficient ansatz to gain reach is exactly what buries its gradients — or to compute the Meyer–Wallach $Q$ of your own circuit's output.

Are expressibility and entangling capability the same thing?

No, but they are tightly correlated. Expressibility measures how uniformly the ansatz covers Hilbert space — the deviation of its second moment from Haar. Entangling capability measures the ensemble-averaged entanglement via the Meyer–Wallach $Q$ . They correlate because reaching Haar-typical states requires near-maximal average entanglement, so a more expressible circuit is usually more entangling. But they are logically distinct: one can build highly entangling circuits that are still poorly expressible — symmetry-restricted circuits that saturate entanglement inside a small subspace — so a designer tracks both numbers, not one.

If high expressibility causes plateaus, why not just use a tiny, inexpressible ansatz?

Because inexpressibility has its own failure mode: an ansatz too restricted to contain the target will train happily to a wrong answer. The product-state ansatz is the extreme — perfectly trainable, entirely useless. The goal is not to minimize expressibility but to match it to the problem: the least expressible circuit whose reachable manifold still includes the solution. This is why chemistry-inspired ansätze encode the target's symmetries (particle number, spin) — shrinking the accessible space to the physically relevant sector keeps gradients alive while retaining the states that matter.

Does a barren plateau just mean my optimizer is bad?

No, and this is the crucial point. A plateau is a property of the landscape, not the optimizer. When $\mathrm{Var}[\partial_\mu C]\sim2^{-n}$ , gradients are exponentially small everywhere in a vast region, so gradient descent, Adam, and natural-gradient methods all stall equally — and gradient-free methods fare no better, since the cost values themselves concentrate exponentially and are indistinguishable within shot noise. The obstruction is an information-theoretic measurement wall: you would need $\sim2^{n}$ shots just to see a descent direction. No optimizer can manufacture signal the measurements cannot resolve; the fix changes the circuit or the cost.

Is the barren-plateau variance always exactly 2 to the minus n?

The clean $2^{-n}$ scaling is the exact- $2$ -design result; real circuits show a family of related decays, and the essential dichotomy is exponential versus polynomial. An approximate $2$ -design gives variance $\Theta(2^{-n})$ ; a shallow, local ansatz with a local cost gives $\Omega(1/\mathrm{poly}(n))$ — decaying only polynomially, hence trainable. Noise- and entanglement-induced plateaus add further exponential decays. The message is qualitative: near a $2$ -design you are exponentially flat, and only structure — locality, symmetry, shallow depth — restores a polynomial floor.

How is expressibility actually measured in practice?

The most common route is the fidelity-sampling method of Sim et al. Draw many pairs of parameter vectors, prepare the output states, estimate their fidelity $F=|\langle\psi_{\boldsymbol\theta}|\psi_{\boldsymbol\theta'}\rangle|^2$ (via a swap test), and histogram the results. Compare that histogram to the analytic Haar law $P_{\mathrm{Haar}}(F)=(N-1)(1-F)^{N-2}$ using the KL divergence — that number is $\mathrm{Expr}$ . It is cheap on a few qubits and serves as an early warning: an ansatz whose small- $n$ histogram already hugs the Haar curve will plateau when scaled up. Frame potentials can also be estimated directly.

Why does the cost observable's locality matter as much as the ansatz?

Because the analysis concerns the pair (ansatz, cost), not the ansatz alone. Cerezo et al. showed that a global cost — an observable coupling all $n$ qubits, such as a projector onto a single target state — induces $2^{-n}$ gradient variance even for a shallow, log-depth circuit nowhere near a $2$ -design. A local cost built from few-qubit observables admits a polynomial gradient floor for shallow circuits. The intuition: a global observable moves only under coordinated changes across the whole register, and such directions are exponentially rare. So trainability is engineered on two fronts — keep the ansatz off the $2$ -design and keep the cost local.