Expressibility of the alternating layered ansatz for quantum computation

1 Introduction

• HEA is often used for hardware implementability, but randomly chosen parameters can make standard cost-function gradients vanish and stall learning.The design challenge is an implementable ansatz that can produce an exact or well-approximating solution after training.Source: §1 · p. 1
• ALT restricts HEA entanglers to local qubits and, under stated conditions, avoids vanishing gradients with a shallow circuit of O(poly(log n)) layers.The cited result assumes a local cost function and 2-design ensembles for circuit blocks.Source: §1 · p. 1
• Because ALT is a subclass of HEA, the paper asks whether its state-generation expressibility is substantially lower, and reports that shallow ALT has the same level as HEA.The authors evaluate expressibility using a measure introduced in prior work.Source: §1 · p. 2
• The paper's stated analysis combines theoretical and numerical comparisons of ALT and HEA expressibility, followed by an examination of how expressibility relates to VQE results.The paper describes these analyses as Sections 3 and 4, respectively.Source: §1 · p. 2

2.1 Indicators of the expressibility

• An ansatz has higher expressibility when its generated-state distribution is closer to the Haar distribution, as measured by A(t)(C).When A(t)(C) is approximately zero, the ansatz can generate almost all states, and generated states are nearly equally distributed.Source: Frame Potential · p. 2
• Lower frame potential and lower KL-divergence-based E(C) indicate higher expressibility by measuring aspects of the generated fidelity distribution against Haar behavior.The frame potential captures moments of fidelity between states generated from randomly sampled circuit parameters; E(C) uses the fidelity distribution.Source: Frame Potential · p. 2, KL-Divergence · p. 3

2.2 Ansatzes

• HEA uses multiple layers of parameterized single-qubit gates and entanglers that connect all qubits.The HEA class is indexed by layer count ℓ and qubit count n.Source: Hardware Efficient Ansatz · p. 3
• ALT uses separated blocks with parameterized single-qubit rotations and fixed within-block entanglers, arranged in alternating partitions across layers.Odd layers use m-qubit blocks; even layers shift the partition, with m/2-qubit end blocks.Source: Alternating Layered Ansatz · p. 3
• The conditions require each Hk to be local, composed of fewer than m neighboring qubits, each block’s unitary ensemble to be a 2-design, and ℓ of order O(poly(log n)); blocks must be deep enough for their unitary distributions to be close to 2-designs.Source: Alternating Layered Ansatz · p. 3
• TEN applies blocks to fixed, non-overlapping groups of m qubits, so its generated states remain products of those block states.The number of blocks per layer is n/m, with n/m assumed to be an integer.Source: Tensor Product Ansatz · pp. 3–4

3 Expressibility of the circuit ansatzes

• Shallow ALT has almost the same expressibility as HEA in frame-potential and KL-divergence measures.The KL-divergence evidence also supports the frame-potential conclusion when ALT’s second frame potential is close to HEA’s.Source: §3 · p. 4, §3.2 · p. 9

3.1 Analytical expression of the frame potential of the ansatzes

• The section derives analytical expressions and upper bounds for the first and second frame potentials of the ansatzes.Its conclusions rely on 2-design assumptions for the relevant unitary ensembles.Source: §3 · p. 4, §3.1 · p. 4
• ALT’s second frame potential is much smaller than TEN’s for both ℓ = 2 and ℓ = 3, indicating higher ALT expressibility.For ALT, increasing layers or block size improves expressibility; at fixed n, using fewer blocks also improves it.Source: The Second Frame Potential · p. 7
• For m = 10, ALT has almost the same expressibility as HEA across the n/m values shown, for both ℓ = 2 and ℓ = 3.The passage says this holds irrespective of total qubit count n.Source: The Second Frame Potential · p. 7, Figure 2 · p. 7
• With m = O(log2 n) and sufficiently random blocks, ALT’s state distribution approaches Haar, and its expressibility approaches HEA’s.The passage states that expressibility and trainability coexist for shallow ALT under this block-size scaling.Source: The Second Frame Potential · p. 8

3.2 Expressibility measured by KL divergence

• The KL-divergence results show that TEN exceeds ALT at fixed n, while increasing layers lowers KL divergence for fixed (m, n).The normalized second frame-potential trend is strongly correlated with the KL-divergence trend.Source: §3.2 · p. 9

4 Application to VQE

• ALT had expressibility as high as HEA according to the KL-divergence comparison.The section identifies the compared circuits as C3,2,4 ALT and HEA.Source: Figure 4 · p. 10, §4 · p. 11
• For Hamiltonian (25), 5 of 100 ALT trajectories reached the ground energy; TEN cannot represent the ground state for any parameter choice, while HEA also did not reach it.Trials used randomly initialized ansatz parameters and Adam optimization with learning rate 0.001.Source: §4 · p. 11, Figure 5 · p. 12
• ALT's average gradient magnitude was larger than HEA's at every evaluated energy, a possible aid against flat landscapes that the authors say needs further study.For ALT at energy −1, the reported average magnitude was approximately 5.6, based on 100 trajectories.Source: §4 · p. 12

5 Conclusion

• Under frame-potential and KL-divergence measures, shallow ALT has almost the same expressibility as HEA.The conclusion states this result when each block entangles O(log n) resource qubits.Source: §5 · p. 12
• A VQE case study suggests that ALT can combine expressibility and trainability.Source: §5 · p. 12
• The formal results cover ℓ=2,3; higher expressibility for larger ℓ is observed numerically, while the conclusion for ℓ≥4 remains conjectural.The rigorous proof for ℓ≥4 is left for future work.Source: §5 · p. 12

• Ua denotes the full circuit unitary, while Ua(i) and Ua(i, j) denote the unitary of layer i and block j in layer i, respectively.Source: A Proof of Theorems · p. 13

A.1 Proof of Theorem 1

• The proof starts from the Haar-measure distribution of the overlap fidelity F between two n-qubit states.Source: §A.1 · p. 13
• Because each block-unitary ensemble is a 2-design, the proof applies the stated integration formula to integrals over those block unitaries.Source: §A.1 · pp. 13–14
• The proof integrates over the blocks in the last layer to obtain the stated intermediate expression, and establishes the other equality by the same method.Source: §A.1 · p. 14

A.2 Proof of Theorem 2

• For final states Ua|0⟩ and Ub|0⟩, the proof writes their overlap as a product of block-unitary contributions raised to the fourth power.Source: §A.2 · p. 14

A.3 Proof of Theorem 3

• The proof states that integrating only the first- and third-layer unitary matrices yields the final formula in the theorem.Source: §A.3 · pp. 14–15
• The displayed computation integrates over unitary matrices from the first and third layers, using the 2-design property for the indicated block ensembles.Source: §A.3 · pp. 15–16
• The resulting expression is identified as the right-hand side of the first equality in equation (33) when n/m = 2.Source: §A.3 · p. 18

A.4 Proof of Theorem 4

• The proof expands a(3,m) and B(3,m) into components of orders O(1) and O(1/2m/2).Source: §A.4 · p. 18
• In g3, each Ri is D or X, and k counts the number of X operators in the sequence.Source: §A.4 · p. 18
• Algorithms evaluate Eqs. (13) and (17) for arbitrary m and derive their expansion formulas computationally.Source: §A.4 · p. 18
• The proof derives an upper bound for F(2)(C3,m,n ALT) using inequalities (49), (50), and (51).Source: §A.4 · p. 19

A.5 Proof of Corollary 1

• The proof treats the inequality for F(2)(C3,m,n ALT).Source: §A.5 · p. 19
• An intermediate inequality is labeled (54).Source: §A.5 · p. 19
• The proof substitutes Eqs. (53) and (54), along with m = 2a log2 n, into Eq. (52).Source: §A.5 · p. 19