Abstract

Deep metric learning has recently shown extremely promising results in the classical data domain, creating well-separated feature spaces. This idea was also adapted to quantum computers via Quantum Metric Learning (QMeL). QMeL consists of a 2 step process with a classical model to compress the data to fit into the limited number of qubits, then train a Parameterized Quantum Circuit (PQC) to create better separation in Hilbert Space. However, on Noisy Intermediate Scale Quantum (NISQ) devices. QMeL solutions result in high circuit width and depth, both of which limit scalability. We propose Quantum Polar Metric Learning (QPMeL ) that uses a classical model to learn the parameters of the polar form of a qubit. We then utilize a shallow PQC with \(R_y\) and \(R_z\) gates to create the state and a trainable layer of \(ZZ(\theta)\)-gates to learn entanglement. The circuit also computes fidelity via a SWAP Test for our proposed Fidelity Triplet Loss function, used to train both classical and quantum components. When compared to QMeL approaches, QPMeL achieves 3X better multi-class separation, while using only 1/2 the number of gates and depth. We also demonstrate that QPMeL outperforms classical networks with similar configurations, presenting a promising avenue for future research on fully classical models with quantum loss functions.

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Figure: QPMeL triplet training loop. The fidelity triplet loss is computed based on the SWAP test fidelity measurement and the gradients are backpropagated throughout. The classical model weights, ZZ parameters and QRC parameters are updated together, having the classical head directly learn the polar coordinates that create separation in Hilbert Space.

Paper Details

\(\newcommand{\ket}[1]{\left|{#1}\right\rangle}\) \(\newcommand{\bra}[1]{\left\langle{#1}\right|}\) \(\bra{\Psi}\Omega\ket{\Psi}\)

\(\newcommand{\braket}[2]{\left\langle{#1}\middle|{#2}\right\rangle}\) \(\braket{\Psi}{\Psi}\) \(\braket{\frac{\Psi}{2}}{\Psi}\)

Math Demo

\(\alpha = \beta\)

Paper Contributions

QPMeL addresses introduces the following ideas:

  1. A novel classical network that encodes classical data into 2 real-valued vectors that are used as Polar coordinates of a qubit. This allows us to utilize the entire 3D space of a qubit, as we are not limited to a single plane.
  2. A hybrid Hilbert space distance metric we dub Fidelity Triplet Loss that measures distance in Hilbert Space while creating the optimization target classically. The distances are measured in-circuit while their difference is computed classically.
  3. Quantum Residual Corrections to speed up model learning and generate more stable gradients by acting as a noise barrier. The parameters absorb noisier gradients to allow the classical model to learn more efficiently.

Quantum State Equation

The final state produced by our Encoding circuit would be: \begin{equation} \ket{\psi} = \bigotimes_{i=0}^n \exp(i\frac{\phi_i}{2}) \cos{\frac{\theta_i}{2}} \ket{0} \; + \; \exp(i \frac{- \phi_i}{2}) \sin{\frac{\theta_i}{2}} \ket{1} \end{equation} where,

\[\begin{align*} \phi_i &= \alpha_k - \alpha_i - \gamma_i \\ k &= (n+i)\mod (n+1) \theta_i &= \theta_{m_i} + \theta_{\Delta_i} \quad | \quad \gamma_i = \gamma_{m_i} + \gamma_{\Delta_i} \\ \theta_{m_i}, \gamma_{m_i} &= f(image,w) \\ \end{align*}\]

Where we have 6 parameters per qubit, 2 from the classical model ($\theta_m,\gamma_m$), 2 learned parameters for the $ZZ$-Gate ($\alpha_i,\alpha_k$) and 2 residuals ($\theta_{\Delta},\gamma_{\Delta}$).

Bibtex

@misc{sharma2023quantum,
      title={Quantum Polar Metric Learning: Efficient Classically Learned Quantum Embeddings}, 
      author={Vinayak Sharma and Aviral Shrivastava},
      year={2023},
      eprint={2312.01655},
      archivePrefix={arXiv},
      primaryClass={quant-ph}
}
Author Details