Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicator

I. INTRODUCTION

• AAE trains a shallow parameterized quantum circuit to approximate ideal data loading, accepting error from limited ansatz expressibility and potentially incomplete optimization.The method is intended for applications that tolerate imperfection in the quantity of interest.Source: §I · p. 1

B. The proposed algorithm

• Unlike the earlier method [40], which approximates amplitude magnitudes, AAE can encode component signs as well as magnitudes.The paper notes that matching magnitudes alone may leave the output with incorrect signs, even at zero cost.Source: §I · p. 2, B. The proposed algorithm · p. 3

I. INTRODUCTION

• AAE combined with quantum SVD approximates stock-market SVD entropy, targeting its global trend rather than precise values.The authors report numerical simulations with real stock-price data as a demonstration.Source: §I · p. 2

A. The goal of the AAE algorithm

• The target is a state whose amplitudes represent the data vector, but exact state generation can require exponentially many gates.AAE instead uses a circuit of O(1) to O(poly(n)) depth, with a fixed post-processing circuit and an auxiliary qubit in the general formulation.Source: A. The goal of the AAE algorithm · p. 2, §I · p. 2

1. Condition for the perfect encoding

• For mixed-sign data, an auxiliary qubit and post-selection after applying a Hadamard transform can yield the target state; the two cases use different loading circuits.In Case 1, U(θ) is used directly; in Case 2, post-processing follows U(θ).Source: §1 · pp. 3–4

2. Optimization of U(θ)

• MMD is used to train model distributions toward target distributions, and the paper uses it for ease of use while noting Stein discrepancy and Sinkhorn divergence as alternatives.The cost compares model and target probability distributions; a characteristic kernel makes zero MMD imply equality of those distributions.Source: §2 · pp. 4–5
• O(Nshot) sample-averaging estimates the MMD with error O(1/√Nshot), avoiding O(N) computation.Samples are obtained by repeatedly measuring the final PQC state.Source: §2 · pp. 4–5
• The cost combines discrepancies for computational-basis and Hadamard-basis measurement distributions in both cases.The training policy minimizes the combined cost; when both discrepancies are zero, the corresponding model distributions match their targets.Source: §2 · p. 5
• The parameter gradients use the parameter-shift rule, and their sampling error depends on Nshot rather than the dimension N.The circuit parameters are updated by gradient descent; the expectation-value estimation error scales as O(1/√Nshot).Source: §2 · p. 5
• Classical shadows may reduce measurement requirements, but extending the analysis to Y-basis measurements and probabilistic coefficient checks is left for future work.The proposed shadow estimation covers Pauli terms formed from I and Z or I and X, with additive error ε using ∝poly(n)/ε^2 measurements.Source: §2 · p. 5

C. Computational complexity of the AAE algorithm

• AAE may face barren plateaus and local minima, and the proposed mitigation approaches require further study.The passage notes that gradients can become exponentially small with qubit count for general highly expressive PQCs and that local minima may require many training trials.Source: C. Computational complexity of the AAE… · p. 6
• Circuit depth tends to increase as the cost threshold is lowered, and the text says further theory is needed to establish whether O(poly(n)) depth suffices for any data-loading circuit.Less precise loading can require smaller depth than near-perfect cost minimization; SVD entropy computation is given as an example.Source: C. Computational complexity of the AAE… · p. 6
• When Nalg is of order poly(log N), AAE has O(N) times smaller execution-stage computational complexity than exact encoding, a benefit for repeated data loading.The table distinguishes training from execution and considers dense and sparse data; exact encoding has no training stage.Source: C. Computational complexity of the AAE… · p. 6, Table I · p. 7
• For dense data, AAE training includes O(N log N) classical computation from the Walsh-Hadamard transform.The text compares this with data registration costs for exact encoding and QRAM, which also require at least O(N) computation.Source: C. Computational complexity of the AAE… · p. 6
• A convex-relaxation complexity bound is conditional on a poly(log N) parameter count, while the relaxation gap remains an open scalability question.The text gives O(poly(log N)poly(1/ε)) gate operations for reaching relaxed cost below ε and leaves the gap's significance for future work.Source: C. Computational complexity of the AAE… · pp. 6–7

D. Some modification on the AAE algorithm

• Replacing the equally weighted MMD terms with a weighted average is proposed as a modification whose effect remains to be investigated.The terms correspond to computational-basis and Hadamard-basis distributions.Source: D. Some modification on the AAE… · p. 7
• Stein discrepancy and Sinkhorn divergence, like MMD, have efficiently computable gradients, whereas KL-divergence requires O(2^n) measurements to compute qθ(j) for each sample.The text therefore says the KL-divergence gradient cannot be efficiently computed in this setting.Source: D. Some modification on the AAE… · p. 7
• Using conditions (4) and (24) instead of condition (5) may reduce the associated classical computation from O(N log N) to O(N).The text says this modified approach may also work, including in Case 2.Source: D. Some modification on the AAE… · p. 8
• A carefully chosen operator X could remove the need for Case 2, but finding one for arbitrary real vectors appears difficult.The proposed two-case approach follows from this difficulty.Source: D. Some modification on the AAE… · p. 8
• The discussed GAN customization uses two quantum generators and two classical discriminators, with the same PQC measured in computational and Hadamard bases.The two generator-discriminator pairs correspond to conditions (4) and (5).Source: D. Some modification on the AAE… · p. 8

III. APPLICATION TO SVD ENTROPY CALCULATION FOR FINANCIAL MARKET INDICATOR

• The section introduces SVD entropy as the quantity considered for stock-price dynamics.Source: §III · p. 8
• The quantum algorithm combines AAE with the variational qSVD algorithm to compute SVD entropy.Source: §III · p. 8
• The section concludes with an in-depth numerical simulation of the proposed application.Source: §III · p. 8

A. SVD Entropy

• Classical computation of SVD entropy requires diagonalizing the N_s × N_s correlation matrix, with complexity O(N_s^3).Source: A. SVD Entropy · p. 9
• SVD entropy is proposed as a financial-crisis indicator: normal periods have large entropy, while crisis-related collective behavior sharpens the eigenvalue distribution and yields relatively small entropy.The passage relates this behavior to collective movement across the market or industrial-sector clusters.Source: A. SVD Entropy · pp. 8–9

B. Computation on quantum devices

• The quantum procedure trains AAE to load normalized logarithmic stock returns, then trains qSVD to diagonalize the stock correlation state and estimates entropy from the circuit output.Source: B. Computation on quantum devices · p. 9, §1 · p. 9, §2 · pp. 9–10

1. The first part: data loading

• The target data state uses n_s + n_t qubits, where n_s = O(log N_s) and n_t = O(log T), yielding an exponential advantage in memory resources.Source: §1 · p. 9

2. The second part: Quantum singular value decomposition

• qSVD variationally transforms the Schmidt basis toward the computational basis by minimizing Hamming distances between measured stock and time bitstrings.The cost is zero when the output has the target Schmidt form, whose paired measurement outcomes are perfectly correlated.Source: §2 · pp. 9–10
• Entropy estimation can be negligible when the post-qSVD state is sparse or T is small; the financial example is expected to be sparse because few large eigenvalues carry important information.The passage notes this is especially the case in abnormal periods, and says the qSVD algorithm targets large eigenvalues.Source: §2 · p. 10

C. Complexity of the algorithm

• AAE uses O(poly(log(N_sT))) quantum gate operations for execution, whereas exact encoding uses O(N_sT) gates and has larger total quantum gate operations.The passage states that both classical computation and exact-data compilation require O(N_sT log N_sT) computation.Source: C. Complexity of the algorithm · p. 10

D. Demonstration

• The dataset covers four Dow Jones stocks using monthly opening prices from April 2008 through March 2009, with five-month windows for each entropy estimate.Each window uses four stocks and five months to form 16 logarithmic-return components.Source: D. Demonstration · p. 11
• AAE plus qSVD well approximates exact SVD entropy, while absolute-value-only loading performs poorly at some terms.The reported AAE estimation errors are within the range considered acceptable for the application, where entropy fluctuates by several percent normally and can change by tens of percent around financial events.Source: Figure 5 · p. 12, D. Demonstration · p. 13
• When L1 and L2 are both below 0.01, mean overlap exceeds 0.94, maximum overlap exceeds 0.96 for every term, and at least 2 of 10 trials meet the condition.Trials with either cost at least 0.01 have relatively small overlap and may yield imprecise qSVD entropy, so the authors say those trials should be discarded.Source: D. Demonstration · p. 13, Table III · p. 13
• The illustrated AAE training distributions closely approximate their targets in both measurement bases, supporting successful SVD-entropy computation.The examples show four terms, and the plotted training run is the best of 10 trials by final cost.Source: D. Demonstration · pp. 14–15, Figure 6 · p. 14
• Entropy is about 0.9 before August 2008, falls to about 0.7 from October through February as stocks co-move, and returns to about 0.9 in March 2009.The authors relate the lower entropy to crisis-period correlation and note that the March rebound is consistent with a cited S&P-index claim that the crisis ended then.Source: D. Demonstration · p. 15
• The authors identify evaluation on other problems and datasets, including an eight-stock example, as important further assessment of AAE.The passage does not report the results of that broader assessment.Source: D. Demonstration · p. 15

• AAE uses a cost composed of two maximum mean discrepancy measures to train the data-loading circuit.Source: §IV · p. 15
• Unlike the previous proposal, AAE can encode the signs of data components rather than only their absolute values.Source: §IV · p. 15
• The algorithm is designed to load classical data into a shallow parameterized quantum circuit.Source: §IV · p. 15

Appendix A: Proof of Theorem 1

• The proof denotes each circuit amplitude by a_j = ⟨j|U(θ)|0⟩ when rewriting the conditions.Source: Appendix A · p. 18
• The argument evaluates the Hadamard-transformed condition at j = 0 as part of establishing the equality condition.Source: Appendix A · p. 18
• In Case 1, the proof concludes that the amplitudes equal the target vector or its negative when the equality condition holds.Source: Appendix A · p. 18

Appendix B: Amplification of the success probability in Case 2

• The ideal Case 2 encoding succeeds with probability 1 after amplitude amplification, rather than 1/2, but requires additional gates.Source: Appendix B · pp. 18–19
• One application of the amplification operator changes the specified amplitude from cos ξ to cos(3ξ); here ξ = π/3, giving −1.Source: Appendix B · p. 19
• The probability-1 result is obtained by ignoring the last two qubits.Source: Appendix B · p. 19

Appendix C: Improvement of AAE

• The appendix considers training with an orthogonal operator X and states that the computational and Hadamard bases are the algorithm’s chosen measurements.Source: Appendix C · p. 19
• The stated conditions (C1) and (C2) yield the target state, while finding a suitable orthogonal operator X for arbitrary data is difficult.Source: Appendix C · pp. 19–20
• A vector and matrix are pair-block-diagonal when row and column permutations transform them into the stated splittable block forms.Source: Appendix C · p. 19
• Because checking pair-block-diagonality for general data is difficult, the appendix says the algorithm instead branches on the values of the data vector.Source: Appendix C · p. 20

Appendix D: Computation of the SVD entropy in the case of eight stocks

• The experiment used monthly opening prices for eight Dow Jones stocks from April 2008 to March 2009, with each entropy computed over a five-month window.The 40 stock-price components yield 32 return components across eight stocks and four time indices; AAE Case 2 uses six qubits including an ancilla.Source: Appendix D · pp. 20–21, Table IV · p. 21
• For qSVD, separate 12-layer ansätze act on the three-qubit stock state and two-qubit time state, with a modified cost for the unequal subsystem sizes.The training objective is minimized to approximate the Schmidt-decomposed state.Source: Appendix D · p. 21
• AAE data-loading circuits used 12 or 13 layers, selected from 20 training trials using the model with the minimum cost.Saved models at iterations 200, 250, 300, 350, and 400 were considered for each trial.Source: Appendix D · p. 21
• With doubled data, Fig. 7 estimation errors were about the same as Fig. 5, with a maximum of about 10% in both; AAE approximated exact SVD entropies, while naive loading performed poorly.Source: Appendix D · p. 22, Figure 7 · p. 22
• The authors infer that AAE depth does not grow exponentially with qubit count, but note optimization issues may worsen as qubits increase and require larger experiments.The larger-size study is explicitly left for future work.Source: Appendix D · p. 22