Rethinking the symmetry-preserving circuits for constrained variational quantum algorithms

We are grateful for your in-time response and we are happy to further discuss with you.

For the discrepancy of the trainability calculation.

Firstly, we would like to be more clear about the difference between our calculation and the results in [1]. We provide a completely different bound on the trainability (comparison shown in the following table). We derivate a whole new bound by integrating all the unitary matrices before and after the $l$-th layer instead of taking the parameters as 0 (as [1] did in their paper). We provide the whole proof and derivation of the bound in the previous reply.

The $Var_\theta[\partial_lC]=\frac{1}{2}(\partial_lC_k)^2$ as shown in Theorem 1 in [1].

Secondly, as for the results in [1]. For the last equation in theorem 2, we have $\partial_lC_k=\langle\psi_k|(\sigma_X^i\otimes\sigma_X^j+\sigma_Y^i\otimes\sigma_Y^j)|\psi_k\rangle=2(\frac{2\tbinom{n-2}{k-1}}{\tbinom{n}{k}})^2=\Omega(\frac{1}{n^6}).$ We have the following three reasons to say why it seems incorrect.

1. If $\partial_lC_k =8(\frac{k(n-k)}{n(n-1)})^2$ is correct in [1], then the usage of Omega notation is incorrect. Big omega notation indicate a larger and equal to, but we don't see under any circumstances that $8(\frac{k(n-k)}{n(n-1)})^2$ is equal to $\frac{1}{n^6}$. $\partial_lC_k=\langle\psi_k|(\sigma_X^i\otimes\sigma_X^j+\sigma_Y^i\otimes\sigma_Y^j)|\psi_k\rangle=2(\frac{2\tbinom{n-2}{k-1}}{\tbinom{n}{k}})^2\neq\Omega(\frac{1}{n^6})$
2. According to our calculation following the scheme in [1], $\partial_lC_k$ should be $4\frac{k(n-k)}{n(n-1)}$ instead of $8(\frac{k(n-k)}{n(n-1)})^2$. This is a clear error. $\partial_lC_k=\langle\psi_k|(\sigma_X^i\otimes\sigma_X^j+\sigma_Y^i\otimes\sigma_Y^j)|\psi_k\rangle\neq2(\frac{2\tbinom{n-2}{k-1}}{\tbinom{n}{k}})^2\neq\Omega(\frac{1}{n^6})$
3. When $k=n/2$, the bound is a constant number without any $n$ in it is clearly incorrect. $\partial_lC_k\neq\langle\psi_k|(\sigma_X^i\otimes\sigma_X^j+\sigma_Y^i\otimes\sigma_Y^j)|\psi_k\rangle\neq2(\frac{2\tbinom{n-2}{k-1}}{\tbinom{n}{k}})^2\neq\Omega(\frac{1}{n^6})$

Thus, we conclude that the results in [1] is incorrect and therefore we provide our own bound based on a completely different scheme.

We can further analyze that if the $k$ is only 1, then $Var_\theta[\partial_lC]\approx \frac{16}{n^3}$. If the $k=\frac{n}{2}$ on the other hand, $Var_\theta[\partial_lC]\approx \binom{n}{n/2}^{-1}$, which is approximate to exponentially small. This result is consistent with the conjecture that the trainability of the circuit is closely related to $d_k$ and smaller $d_k$ will lead to better trainability. We will make sure a detailed version of the analysis on the trainability will be included in the revision.

2. We would like to further restate the contributions of our paper and hope this can dissolve your concern.

1. We revisit the existing works on HW preserving and point out the limitations of these works.
2. We connect various theories to guide the design of HW preserving ansatz. We first utilize the quantum optimal control theory to analyze the capability and expressivity of HW preserving ansatz to see whether the answer to a problem is within the reachable space of the ansatz. We then provide analysis on the trainability of the HW preserving ansatz to see whether we can find a path to reach the answer if it is guaranteed to be in the reachable space by the previous step. Finally, we utilize the overparameterization theory to indicate the approximate number of layers we need to solve the problem.
3. We conducted detailed numerical experiments on constrained VQAs, which are better test cases than [1]. Two tasks involve symmetry-preserving ground state energy estimation (well-studied case) and feature selection (well-studied and investigated by DWave [2]).

To the best of our knowledge, we are the first to combine so many theories to provide a thorough analysis of a quantum ansatz. Compared to previous HW preserving papers such as [1], we rectify the theoretical analysis and the utility of the HW preserving ansatz beyond their currently limited applications. Compared to previous works on hard constrained QAOAs [3], we provide a tool for constructing and analyzing the whole group of HW preserving ansatz instead of sieving through existing gates to seek HW preserving properties and use numerical results to decide how to design circuits.

To sum up, HW preserving is not a newly proposed idea, but it is never thoroughly analyzed and is often used on unsuitable problems. Some of the theoretical results might have been proposed before but it is novel to combine those theories together for one specific ansatz. We never know why certain HW preserving gates can not solve a given problem and whether the bad results are caused by the fact that the answer is unreachable or poor trainability of the ansatz or lack of layers. We strongly believe that this work can help the QML community by improving the understanding of the HW preserving ansatz. [4] is also a paper that aims to improve the understanding of the community by mostly reviewing the existing ideas and is accepted as a spotlight paper in NeurIPS24. We hope that this work can also be valued and attract more attention to HW preserving circuits.

[1] Cherrat, El Amine, et al. "Quantum Deep Hedging." arXiv preprint arXiv:2303.16585 (2023).

[2] https://github.com/dwave-examples/feature-selection-notebook/blob/master/01-feature-selection.ipynb

[3] Hadfield, Stuart, et al. "From the quantum approximate optimization algorithm to a quantum alternating operator ansatz." Algorithms 12.2 (2019): 34.

[4] Abbas, Amira, et al. "On quantum backpropagation, information reuse, and cheating measurement collapse." arXiv preprint arXiv:2305.13362 (2023).

W3: I think this idea of preserving the Hamming weight is not very new, and the theoretical results are not very surprising to me. For this reason, I feel the contribution of this work is not so large, although the proposed ansatz appear to perform better than existing ones.

Q3: Throughout the paper, it is stated that this is a "revisiting" of the Hamming weight preserving ansatz. What is the status of the previous work using these ansatz, and what are the specific improvements made in this paper compared to previous proposals?

Ans: Thanks for your comments. We will answer the above two questions and weaknesses together.

As suggested by Reviewer ADjN, we have provided an analysis on the trainability of HW preserving ansatz (see the answer for weakness 3 for ADjN for detailed derivation). We initially omit the analysis on the trainability since [1] has provided one derivation. However, after careful review, we believe the results in [1] are incorrect and thus we provide our own derivation. We would like to restate the contributions of this work:

1. We revisit the existing works on HW preserving and point out the limitations of these works.
2. We connect various theories to guide the design of HW preserving ansatz. We first utilize the quantum optimal control theory to analyze the capability and expressivity of HW preserving ansatz to see whether the answer to a problem is within the reachable space of the ansatz. We then provide analysis on the trainability of the HW preserving ansatz to see whether we can find a path to reach the answer if it is guaranteed to be in the reachable space by the previous step. Finally, we utilize the overparameterization theory to indicate the approximate number of layers we need to solve the problem.
3. We conducted detailed numerical experiments on constrained VQAs, which are better test cases than [1]. Two tasks involve symmetry-preserving ground state energy estimation (well-studied case) and feature selection (well-studied and interested by DWave [2]).

To the best of our knowledge, we are the first to combine so many theories to provide a thorough analysis of a quantum ansatz. Compared to previous HW preserving papers such as [1], we rectify the theoretical analysis and the utility of the HW preserving ansatz beyond their currently limited applications. Compared to previous works on hard constrained QAOAs [3], we provide a tool for constructing and analyzing the whole group of HW preserving ansatz instead of sieving through existing gates to seek HW preserving properties and use numerical results to decide how to design circuits.

To sum up, HW preserving is not new, but it is never thoroughly analyzed and is often used on unsuitable problems. The theoretical results might not be surprising to you but we never know why certain gates can not solve a given problem and whether the bad results are caused by the fact that the answer is unreachable or poor trainability of the ansatz or lack of layers. We strongly believe that this work can help the QML community by improving the understanding of the HW preserving ansatz. [4] is also a paper that aims to improve the understanding of the community by mostly reviewing the ideas and is accepted as a spotlight paper in NeurIPS24. We hope that this work can also be valued and attract more attention to HW preserving circuits.

[1] Cherrat, El Amine, et al. "Quantum Deep Hedging." arXiv preprint arXiv:2303.16585 (2023).

[2] https://github.com/dwave-examples/feature-selection-notebook/blob/master/01-feature-selection.ipynb

[3] Hadfield, Stuart, et al. "From the quantum approximate optimization algorithm to a quantum alternating operator ansatz." Algorithms 12.2 (2019): 34.

[4] Abbas, Amira, et al. "On quantum backpropagation, information reuse, and cheating measurement collapse." arXiv preprint arXiv:2305.13362 (2023).

W4: Some typos in the paper

Ans: Thanks for your careful review and we will correct these typos in the revision.

W3: More discussion on the trainability is required.

Ans: Thanks for your thoughtful advice. We carefully reviewed the results and proof in [1] after seeing your comments and we are surprised to find it incorrect. There are still chances that we misunderstood the calculation for the variance bound in [1], so we decided to give our results here with a sketch proof. Detailed derivation will be included in the revision.

We first discuss the results shown in [1]. We tried the derivation of theorem 2 and find that $\partial_lC_k=\langle\psi_0|(\sigma_X^i\otimes\sigma_X^j+\sigma_Y^i\otimes\sigma_Y^j)|\psi_0\rangle$ should be $1$ instead of $\frac{1}{4}$. For the initial state in the HW k subspace, the results in [1] are $\partial_lC_k=\langle\psi_k|(\sigma_X^i\otimes\sigma_X^j+\sigma_Y^i\otimes\sigma_Y^j)|\psi_k\rangle=2(\frac{2\tbinom{n-2}{k-1}}{\tbinom{n}{k}})^2=\Omega(\frac{1}{n^6})$, where according to our derivation, the results should be $\partial_lC_k=\langle\psi_k|(\sigma_X^i\otimes\sigma_X^j+\sigma_Y^i\otimes\sigma_Y^j)|\psi_k\rangle=2\frac{2\tbinom{n-2}{k-1}}{\tbinom{n}{k}}$, $\frac{\tbinom{n-2}{k-1}}{\tbinom{n}{k}}=\frac{k(n-k)}{n(n-1)}$. Even if the original result is correct that $\partial_lC_k=8(\frac{k(n-k)}{n(n-1)})^2$, $\partial_lC_k=\mathcal{O}(1)$ if we take $k=n/2$, which seems incorrect.

Now we provide the results based on our derivation. Without loss of generality, we take RBS gate as an example since [1] also derives the bound based on RBS gate.

Theorem 1. Consider a quantum circuit operating in the subspace with Hamming Weight $k$. The variance of the cost function partial derivative is $Var_\theta[\partial_lC]\approx \frac{16k^2(n-k)^2}{n^4d_k}$

Proof. Consider the partial derivative of the cost function $C$ with respect to the parameters $\theta$. For some parameter $\theta_l$ in the $l$-th RBS gate, we have: $\partial_{l} C(\theta)=\partial_l(Tr[U(\theta)\rho U(\theta)^{\dagger}O])$ $=\partial_l(Tr[U_-\rho U_-^{\dagger} O_+])$ $=iTr[U_-\rho U_-^{\dagger} [H_l,O_+]],$

where $\rho$ is the input state, $O$ is the observable to measure, $U_-$ denotes the unitary matrix of the circuit before the $l$-th gate and $U_+$ denotes the unitary matrix after gate $l$. $O_+=U_+^\dagger OU_+$. The variance of the partial derivative is $Var_{\theta}[\partial_l C]=\int_{U_+}dU_+ \int_{{U_-}}dU_- (\partial_l C(\theta))^2$ $=\int_{{U_+}}dU_+ \int_{{U_-}}dU_- (iTr[U_-\rho U_-^{\dagger} [H_l,O_+]])^2,$ where $[\cdot,\cdot]$ denotes the commutator of two matrices. $Var_\theta[\partial_lC]=-\int_{{U_+}}dU_+ (\frac{Tr[\rho^2]Tr[[H_l,O_+]^2]}{d_k^2-1}-\frac{Tr^2[\rho]Tr[[H_l,O_+]^2]}{d_k(d_k^2-1)})$ $=-\int_{U_+}dU_+ (Tr[ [H_l,O_+]^2] \frac{d_k\times Tr[\rho^2]-Tr^2[\rho]}{d_k(d_k^2-1)})$ $=-\frac{d_k\times Tr[\rho^2]-Tr^2[\rho]}{d_k(d_k^2-1)}\int_{{U_+}}dU_+ Tr[ [H_l,O_+]^2] .$ The initial state $|\psi_0\rangle$ is set to be the uniform superposition over all computational basis in the $d_k$ subspace. Thus, we have $Tr[\rho]=1,Tr[\rho^2]=1$. $Var_\theta[\partial_lC]=-\frac{1}{d_k(d_k+1)}\int_{\mathcal{U_+}}dU_+ Tr[ [H_l,O_+]^2]$ $=-\frac{2}{d_k(d_k+1)}[Tr(H_lO_+H_lO_+)-Tr(H_lH_lO_+O_+)]$ $=-\frac{2}{d_k(d_k+1)}(\frac{Tr[H^2_l]Tr^2[O]}{d_k^2-1}-\frac{Tr[H^2_l]Tr[O^2]}{d_k(d_k^2-1)}-\frac{Tr[H^2_l]Tr[O^2]}{d_k})$ $=-\frac{2Tr[H_l^2]}{d_k(d_k+1)}(\frac{Tr^2[O]-d_k Tr[O^2]}{d_k^2-1}),$ where $Tr[H_l^2]=2*\binom{n-2}{k-1}=\frac{2k(n-k)}{n(n-1)}d_k$. We take $Z_0$ as the observable, then $Tr[O]=\frac{d_k(n-2k)}{n}, Tr[O^2]=d_k$ (other observables will also hold with the same magnitude). Thus, we have $Var_{\theta}[\partial_l C]=-\frac{2}{d_k(d_k+1)}\times\frac{2k(n-k)d_k}{n(n-1)}\times(\frac{\frac{d_k^2(n-2k)^2}{n^2}-d_k^2}{d_k^2-1})$ $=\frac{4k(n-k)}{(d_k+1)n(n-1)}\times\frac{d_k^2(n^2-(n-2k)^2)}{(d_k^2-1)n^2}$ $=\frac{4k(n-k)}{(d_k+1)n(n-1)}\times\frac{d_k^2(4nk-4k^2)}{(d_k^2-1)n^2}$ $=\frac{16k^2(n-k)^2 d_k^2}{(d_k+1)n^3(n-1)(d_k^2-1)}\approx \frac{16 k^2(n-k)^2}{n^4 d_k}.$ $\blacksquare$

We can further analyze that if the $k$ is only 1, then $Var_\theta[\partial_lC]\approx \frac{16}{n^3}$. If the $k=\frac{n}{2}$ on the other hand, $Var_\theta[\partial_lC]\approx \binom{n}{n/2}^{-1}$, which is approximate to exponentially small. This result is consistent with the conjecture that the trainability of the circuit is closely related to $d_k$ and smaller $d_k$ will lead to better trainability. We will make sure a detailed version of the analysis on the trainability will be included in the revision.

[1] Cherrat, El Amine, et al. "Quantum Deep Hedging." arXiv preprint arXiv:2303.16585 (2023).

We are truly grateful for your appreciation of our work, and we also thank you for all the thoughtful and valuable suggestions to improve the manuscript! It is really nice to exchange ideas with you.