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On the practical usefulness of the Hardware Efficient Ansatz

Lorenzo Leone, Salvatore F. E. Oliviero, Lukasz Cincio, M. Cerezo

arXiv:2211.01477v2quant-ph

TL;DR

The practical usefulness of HEAs for particular tasks had not been rigorously established. The paper analyzes when shallow HEAs are trainable or untrainable based on input entanglement. It finds volume-law inputs lead to untrainability, while area-law inputs support trainable QML settings and a possible route to quantum advantage.

  • Problem

    The practical usefulness of HEAs for particular tasks remained unclear, with rigorous guidance on where shallow HEAs are suitable still lacking.

    Source: §1 · p. 1, §9 · p. 13

  • Method

    The paper establishes rigorous results relating shallow-HEA trainability to input-state entanglement and analyzes example QML tasks.

    Source: §9 · pp. 13–14

  • Results

    Volume-law inputs make HEAs untrainable through deterministic cost concentration, whereas area-law inputs yield trainability and loss-function anti-concentration.

    Source: §9 · p. 13

  • Takeaways & Limitations

    Area-law QML inputs are a potential regime for shallow-HEA quantum advantage, but classical-data embeddings must not themselves be classically simulable.

    Source: §9 · p. 14

    The volume-law exclusion applies to unstructured HEAs; structured, problem-aware HEAs may avoid exponential concentration when they use prior knowledge of input states.

    Source: §9 · p. 14

Abstract

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Variational Quantum Algorithms (VQAs) and Quantum Machine Learning (QML) models train a parametrized quantum circuit to solve a given learning task. The success of these algorithms greatly hinges on appropriately choosing an ansatz for the quantum circuit. Perhaps one of the most famous ansatzes is the one-dimensional layered Hardware Efficient Ansatz (HEA), which seeks to minimize the effect of hardware noise by using native gates and connectives. The use of this HEA has generated a certain ambivalence arising from the fact that while it suffers from barren plateaus at long depths, it can also avoid them at shallow ones. In this work, we attempt to determine whether one should, or should not, use a HEA. We rigorously identify scenarios where shallow HEAs should likely be avoided (e.g., VQA or QML tasks with data satisfying a volume law of entanglement). More importantly, we identify a Goldilocks scenario where shallow HEAs could achieve a quantum speedup: QML tasks with data satisfying an area law of entanglement. We provide examples for such scenario (such as Gaussian diagonal ensemble random Hamiltonian discrimination), and we show that in these cases a shallow HEA is always trainable and that there exists an anti-concentration of loss function values. Our work highlights the crucial role that input states play in the trainability of a parametrized quantum circuit, a phenomenon that is verified in our numerics.

1 Introduction

Although HEAs are widely used and can be trainable when shallow, their practical usefulness for particular tasks had not been rigorously established. This work identifies contexts where they should likely be avoided and an area-law QML setting where shallow HEAs are trainable.

Introduction

  • HEAs use hardware-native gates to reduce hardware-noise effects, but their broad applicability may limit performance across tasks.Deep HEAs can suffer barren plateaus, while shallow HEAs can avoid them.Source: §1 · p. 2
  • The paper addresses the unresolved question of which problems are suitable for HEAs, whose practical usefulness had remained unclear.HEAs were commonly used irrespective of task, despite calls for problem-specific ansatzes.Source: §1 · p. 2
  • The paper identifies product-state VQA tasks as settings where HEAs should likely be avoided because the resulting algorithm can be efficiently classically simulated.It also proves that QML with volume-law input data should not use HEAs, connecting input entanglement to cost concentration and barren plateaus.Source: §1 · p. 2
  • For QML with area-law input data, shallow HEAs are trainable and may offer a route to quantum advantage.The paper guarantees the absence of barren plateaus in this setting and highlights input states as crucial to trainability.Source: §1 · p. 2

2 Framework

VQAs and QML models both train parametrized quantum circuits, but they differ in their tasks and input states. The paper studies a one-dimensional alternating-layer HEA that follows hardware connectivity and uses native gates.

2.1 Variational Quantum Algorithms and Quantum Machine Learning

  • VQAs use a parametrized circuit to prepare an operator's ground state from a fiduciary state by minimizing a cost function.The cost function is minimized over the circuit parameters.Source: §2.1 · p. 2
  • QML models train a parametrized circuit to predict dataset labels by minimizing a loss function.The input states come from the dataset and may be complex quantum states.Source: §2.1 · pp. 2–3
  • Both VQAs and QML models train a parametrized circuit with a classical optimizer, but VQAs usually use easy-to-prepare inputs while QML uses dataset states.The frameworks differ in whether the task is problem-driven or data-driven.Source: §2.1 · p. 3, Figure 1 · p. 3

2.2 Hardware Efficient Ansatz

  • The HEA uses native gates and generators selected according to a device's connectivity and interactions to reduce circuit depth.The HEA is a generic name for ansatzes designed around a device's native gate alphabet.Source: §2.2 · p. 3
  • The paper studies a one-dimensional alternating-layer HEA with nearest-neighbor gates arranged in a brick-like pattern.Its depth D is the number of layers, which use the hardware's native connectivity to maximize operations and avoid idle qubits.Source: §2.2 · p. 3

3 Trainability of the HEA

Barren plateaus make optimization landscapes exponentially flat and can require exponentially many measurement shots. The section reviews depth and measurement conditions for trainability, then shows that input-state marginals also matter.

3.1 Review of the literature

  • A barren plateau is an exponentially flat optimization landscape that can require exponentially many measurement shots to resolve a cost-minimizing direction.Such shot complexity can erase a potential quantum advantage.Source: §3.1 · p. 4
  • For polynomial-depth HEAs, noise can cause deterministic cost concentration, while high expressibility can cause probabilistic concentration.Earlier results establish that shallow depth and local measurements are necessary conditions for avoiding barren plateaus.Source: §3.1 · p. 4
  • With depth D in O(log(n)), local measurements yield gradients that vanish only polynomially, whereas global measurements lead to barren plateaus.This contrast is stated for the reviewed HEA trainability conditions.Source: §3.1 · p. 4

3.2 A new source for untrainability

  • The gradient-variance lower bound depends on reduced input states within the measurement operator's light-cone, whose support grows by at most 2D qubits.The section identifies depth D in O(log(n)) as a necessary trainability condition.Source: §3.2 · p. 5, Figure 2 · p. 5
  • Even shallow, local HEAs can lose trainability when input-state marginals or the measurement operator are exponentially close to the identity.In that case, the gradient lower bound becomes trivial.Source: §3.2 · p. 5
  • Trainability therefore depends not only on circuit depth and measurement locality but also on the input state and the measured operator.The paper notes that practical applications need to avoid operators with exponentially vanishing non-identity Pauli support.Source: §3.2 · p. 5

4 Entanglement and information scrambling

The paper relates volume-law entanglement in small subsystems to information scrambling and contrasts it with area-law states. It also introduces an information measure for how distinguishable a subsystem is from the maximally mixed state.

4 Entanglement and information scrambling

  • The paper's area-law and volume-law definitions are geometry-agnostic, although the authors state that this distinction is relevant to their analysis.The definitions use the entropy of entanglement across a bipartition.Source: §4 · p. 6
  • The information measure quantifies the maximum distinguishability between a subsystem's reduced state and the maximally mixed state.When this measure is exponentially small, a local measurement on that subsystem cannot efficiently reveal information in the input state.Source: §4 · p. 6
  • Under the paper's definitions, volume-law entanglement across bipartitions with O(log(n))-sized subsystems implies exponentially suppressed information in those subsystems.The paper uses this relation to define information scrambling.Source: §4 · p. 6

5 HEA and the role of entanglement

The section distinguishes shallow HEAs on volume-law inputs, which are untrainable, from area-law settings where trainability is guaranteed. Whether the latter offers a quantum advantage depends on classical simulability, which the paper does not settle in general.

5.1 HEA and volume law of entanglement

  • Shallow HEAs are untrainable on volume-law inputs, with the toy model showing deterministic exponential concentration of cost values.Source: §5.1 · p. 6, §5.1.1 · p. 7

5.1.1 A toy model

  • A shallow circuit’s backward-evolved local observable acts on at most 2D qubits; for highly entangled inputs, tracing out the rest yields concentration.Source: §5.1.1 · pp. 6–7

5.1.2 Formal statement

  • For logarithmic-depth circuits and logarithmic-locality observables, trainability depends on input information scrambling; excessive scrambling prevents local measurements from accessing information.Source: §5.1.2 · p. 7
  • Haar-random inputs concentrate with overwhelming probability, and this remains true when the shallow HEA processes two copies of the state.Source: §5.1.2 · pp. 7–8

5.2 HEA and area law of entanglement

  • For area-law inputs, shallow HEAs under the stated gate and measurement assumptions anti-concentrate, and the paper links this to the absence of barren plateaus.Source: §5.2 · p. 8
  • Area-law inputs with polynomial-bond-dimension MPS descriptions can be efficiently simulated, while establishing quantum advantage for these or other cases is beyond the paper’s scope.Source: §5.2 · pp. 8–9

6 Implications of our results

The implications depend on the input states and the task. Shallow HEAs are generally poor candidates for VQA quantum advantage and should be avoided for QML volume-law data, while non-classically-simulable area-law QML data remain a potential use case.

6.1 Implications to VQAs

  • In common VQA examples, separable initial states are trainable but classically simulable, so shallow HEAs should likely be avoided when seeking quantum advantage.Source: §6.1 · p. 9

6.2 Implications to QML

  • Figure 4 contrasts trainable QML tasks on area-law datasets with untrainable tasks on volume-law datasets; the area-law case may also offer quantum advantage.Source: Figure 4 · p. 10
  • For QML, the input dataset determines HEA usability: avoid volume-law data and efficiently classically described area-law data, while non-classically-simulable area-law data remain a potential use case.Source: Figure 3 · p. 9, §6.2 · p. 10

7 Random Hamiltonian discrimination

The section frames Hamiltonian discrimination as distinguishing states evolved under a symmetry-preserving Hamiltonian from those evolved under a general one. For GDE Hamiltonians, long evolution leads to loss concentration, whereas short-time area-law inputs yield anti-concentration and a setting where the QML task can succeed.

7.1 General framework

  • The dataset pairs states evolved from symmetry-sector vectors under a symmetry-preserving Hamiltonian with states evolved under a general Hamiltonian, which the model must distinguish.The task requires a parameter setting whose output is 1 for the symmetry-preserving class and not close to 1 for the general-Hamiltonian class.Source: §7.1 · pp. 10–11

7.2 Gaussian Diagonal Ensemble Hamiltonian discrimination

  • The example uses Gaussian Diagonal Ensemble Hamiltonians, built from random energies and projectors onto random Haar states, to study how evolution time affects entanglement and task success.The section presents GDE as an analytically tractable example and notes that broader random-Hamiltonian ensembles are possible extensions.Source: §7.2 · p. 11
  • Beyond t ∼ √n, the loss concentrates with overwhelming probability, yielding a no-go result for the QML task as entanglement grows with evolution time.The concentration bound’s dependence on t reflects the monotonically increasing entanglement reported for random GDE Hamiltonians.Source: §7.2 · pp. 11–12
  • At t = O(log(n)), GDE-evolved states from factorized inputs satisfy an area law and the loss anti-concentrates, supporting success on the Hamiltonian-discrimination task.The paper presents this regime as a potential quantum-speedup example because random Hamiltonian evolution is classically difficult to simulate.Source: §7.2 · pp. 11–12

8 Numerical simulations

The simulations vary the evolution time used to prepare HEA inputs and examine gradient norms and reduced-state entropy. Gradients decline as evolution time and entanglement increase, while their saturation value decreases polynomially with system size.

8 Numerical simulations

  • The simulations use time-evolved random product states under a nearest-neighbor Heisenberg model as inputs to a shallow HEA with a sum of local Z measurements.The HEA has an initial layer of general single-qubit rotations followed by two-qubit gates on alternating pairs.Source: §8 · pp. 12–13
  • At long evolution times, the gradient norm reaches a saturation value that decays polynomially with the number of qubits.The simulations relate the saturation regime to increased entanglement in the time-evolved input states.Source: §8 · p. 13
  • For small evolution times, gradients remain large independently of system size, then decrease as evolution time increases toward saturation.At t → 0, the input is a product state; with the stated local measurements and HEA structure, gradient norms are independent of n.Source: §8 · p. 13
  • At saturation, reduced-state entropy is positively correlated with gradient decay, consistent with more entangled inputs having smaller gradients and more concentrated landscapes.The entropy measure is computed on two neighboring qubits.Source: §8 · p. 13

9 Discussion and conclusions

The paper identifies input-state entanglement as a determinant of shallow HEA trainability: volume-law QML inputs are untrainable, while area-law inputs yield trainable landscapes. It connects these regimes to classical simulability and gives a Hamiltonian-discrimination example within an area-law time window.

9 Discussion and conclusions

  • Volume-law input states make HEA QML untrainable through deterministic cost concentration, whereas area-law inputs yield trainability and loss anti-concentration.For area-law inputs, the loss differs at least polynomially at sufficiently different parameter settings.Source: §9 · p. 13
  • The identified untrainability source is input-state entanglement, distinct from prior barren plateaus caused by volume-law entanglement generated by deep circuits.The shallow HEA itself cannot create volume-law entanglement, yet volume-law input states can still make it untrainable.Source: §9 · p. 13
  • Shallow HEAs with polynomial bond-dimension inputs admit polynomial-overhead classical simulation, making typical product-state VQA inputs unsuitable for quantum speedups.For QML, area-law states remain a candidate window because the portion admitting efficient classical descriptions is exponentially small, though this does not guarantee advantage.Source: §9 · p. 14
  • Hamiltonian discrimination provides an example: area-law entanglement and absence of barren plateaus hold within a time window scaling logarithmically with qubit count.Longer evolution makes the task less feasible as entanglement grows.Source: §9 · p. 14
  • The proposed recipe uses entangled quantum inputs, a shallow circuit, and local measurements; classical-data use requires embeddings that yield area-law states not classically simulable.The authors identify such entangled-but-tamed embeddings as a research direction.Source: §9 · p. 14
  • The volume-law exclusion applies to unstructured HEAs; problem-aware ansatzes may avoid exponential concentration when they use prior knowledge of input states.The paper points to symmetry-aware geometric quantum machine learning as an example of structured, problem-dependent ansatz design [35] [36] [39].Source: §9 · p. 14

A.1 Proof of Theorem 1

This appendix proof bounds cost deviation using information scrambling and Pauli-operator support, then derives concentration results under logarithmic-size locality and depth conditions.

A.1 Proof of Theorem 1

  • Theorem 1’s proof decomposes the measurement into Pauli terms and bounds deviation from the trivial cost using information-theoretic information scrambling.The proof groups nearby qubits into clusters to bound the supports involved.Source: §A.1 · pp. 19–20

A.2 Proof of Corollary 1

  • When information is scrambled and measured Pauli supports are O(log(n)), Corollary 1 derives an exponentially small cost deviation.The argument bounds information scrambling on the relevant regions before applying the support-size condition.Source: §A.2 · pp. 20–21

A.3 Proof of Corollary 2

  • For Haar-random inputs, Corollary 2 uses Levy’s lemma to bound deviation from trivial cost with overwhelming probability when Pauli supports and circuit depth are O(log(n)).The proof applies the bound to measurements composed of polynomially many local Pauli terms.Source: §A.3 · pp. 21–22

A.4 Proof of Corollary 3

The proof of Corollary 3 analyzes reduced states of Haar-random inputs through their multi-copy moments, including a two-copy swap calculation, and invokes Levy’s lemma for typicality bounds.

A.4 Proof of Corollary 3

  • For a Haar-random n-qubit state, the proof studies k copies and the partial trace of their symmetrized permutation operator over the complement of a subsystem.It denotes the Hilbert-space dimension by d = 2^n.Source: §A.4 · pp. 22–23
  • For k = 2, the proof uses a swap operator and obtains a deviation term of order O(d^-1).The calculation traces swap components over the subsystem and its complement.Source: §A.4 · p. 23
  • For general k, the proof assumes subsystem symmetry across copies and applies Levy’s lemma to obtain typicality bounds for epsilon greater than O(2^-n).The Lipschitz argument bounds expectation values on the multi-copy state.Source: §A.4 · pp. 22–24

B.1 Proof of Theorem 2

The proof treats shallow-HEA trainability under area-law inputs and relates absence of barren plateaus to anti-concentration. Its converse argument studies cost differences between parameter settings separated by inverse-polynomial distances.

B.1 Proof of Theorem 2

  • The proof connects parameter settings by polynomially many single-coordinate changes, with at least one shift of inverse-polynomial size.It uses the polynomial closeness of the two settings and the polynomial number of parameters.Source: §B.1 · p. 24
  • Theorem 2 considers logarithmic-depth shallow HEAs with local two-qubit 2-design gates and polynomially many local traceless Pauli measurements; its proof concerns area-law input states.The supplied proof states that Theorem 2’s absence-of-barren-plateaus result implies cost anti-concentration.Source: §B.1 · p. 25, §B.2 · p. 25

B.2 Proof of Proposition 1

  • Proposition 1 proves the converse implication, defining anti-concentration through polynomially separated costs at parameter settings differing by at least an inverse-polynomial shift.The proof invokes the parameter-shift rule to relate cost differences to gradients.Source: §B.2 · p. 25

C Isospectral twirling and proof of Proposition 3,Thereom 3 and Proposition 4

The appendix uses isospectral twirling and GDE spectral averages to establish purity-fluctuation and anti-concentration bounds. These proofs reduce the loss-concentration question to subsystem-purity calculations.

C.1 Isospectral twirling

  • Isospectral twirling averages unitary evolution over eigenvectors, followed by a spectral average; the paper uses GDE spectra because their spectral form factors are readily computed.The section introduces this framework for the subsequent proofs.Source: §C.1 · p. 26

C.2 Proof of Proposition 3

  • The Proposition 3 proof evaluates the loss average by isospectral twirling and uses the symmetry eigenstate condition to recover the stated expression.The input state is factorized, and the chosen observable at θ0 is the symmetry S.Source: §C.2 · p. 27

C.3 Proof of Theorem 3

  • Theorem 3’s concentration proof applies Jensen’s and Markov’s inequalities to the twirled loss, under a product input across A and B with |A| ≪ |B|.The argument also uses Tr[O] = 0.Source: §C.3 · p. 27

C.4.1 An anti-concentration inequality

  • The proof reduces anti-concentration of loss values to anti-concentration of subsystem purity for a subsystem of O(log(n)) qubits.It uses Cantelli’s inequality and bounds on the purity’s first and second moments.Source: §C.4.1 · p. 28

C.4.2 Computing PGDE

  • Haar invariance lets the average purity be computed using an averaged product state across Λ and its complement.This is the calculation of PGDE for GDE Hamiltonians.Source: §C.4.2 · p. 28

C.4.3 Computing QGDE

  • The second purity moment is computed by rewriting its terms as high-order OTOCs and using an asymptotic spectral-form-factor formula.The direct eighth-fold averaging involves 8! permutations, motivating the OTOC-based calculation.Source: §C.4.3 · pp. 29–30

C.4.4 Final bound

  • For the GDE, isospectral twirling yields purity fluctuations that are then used to obtain the anti-concentration bound.The proof first bounds the purity fluctuations and then applies the earlier inequality.Source: §C.4.4 · p. 30
  • The final bound combines purity fluctuations with an anti-concentration inequality, choosing θ = O(d^1/4) for the stated bound.The fluctuations are evaluated for GDE Hamiltonians before the theorem is concluded.Source: §C.4.4 · p. 30

D Useful Lemmas

These lemmas analyze how shallow hardware-efficient circuits spread operator support and how gate averaging controls loss and gradient calculations. One lemma applies under an area-law input-state condition.

Support growth

  • Repeated local gate layers expand an operator’s support, while the proof bounds the evolved measurement support using the clusters of its terms.The support can grow by a factor m at each layer; the cluster argument relates term separation to mD.Source: D Useful Lemmas · p. 32

Asymptotic qualification

  • The stated result is asymptotic; for small n, n may itself belong to O(poly(log(n))).The authors flag this as a finite-size qualification to the result.Source: D Useful Lemmas · p. 32

Cost-function averaging

  • The tensor-network calculation integrates successive two-qubit gates and uses partial traces to evaluate the cost-function average.The figure depicts averaging over Vμ and Vμ′ when they form 2-designs.Source: D Useful Lemmas · p. 33, Figure 9 · p. 33

Gradient-product calculation

  • For gradient products associated with two parameterized gates, the proof evaluates gate averages using local 2-design assumptions and integrations over the gates.The gates’ relative positions divide the circuit into portions before, between, and after them.Source: D Useful Lemmas · pp. 33–34

Lemma 4

  • For shallow HEAs with area-law input states, Lemma 4 gives a bound for measurements built from polynomially many traceless Pauli terms on neighboring qubits.The displayed bound’s expression is not included in these passages.Source: D Useful Lemmas · p. 36

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