Data compression for quantum machine learning

The advent of noisy-intermediate scale quantum computers has introduced the exciting possibility of achieving quantum speedups in machine learning tasks. These devices, however, are composed of a small number of qubits and can faithfully run only short circuits. This puts many proposed approaches for quantum machine learning beyond currently available devices. We address the problem of compressing classical data into efficient representations on quantum devices. Our proposed methods allow both the required number of qubits and depth of the quantum circuit to be tuned. We achieve this by using a correspondence between matrix-product states and quantum circuits and further propose a hardware-efficient quantum circuit approach, which we benchmark on the Fashion-MNIST dataset. Finally, we demonstrate that a quantum circuit-based classifier can achieve competitive accuracy with current tensor learning methods using only 11 qubits.

Figure 1

(a) Our encoding scheme consists of splitting the image into patches, each of which is encoded in a quantum superposition state consisting of address qubits and a color qubit, as defined in Eq. (1). (b) In the MPS learning protocol, each patch is separately encoded into an MPS consisting of tensors with physical dimension 2 corresponding to address (circle) qubits and the color (square) qubit. These MPS are concatenated and contracted with an MPS-based classifier (purple) with fixed bond dimension. An additional dimension 10 classifer leg, represented by three lines, is used to classify the Fashion-MNIST images. (c) An example encoded Fashion-MNIST image with bond dimension ${\chi }_{\text{img}}=4$ for different numbers of patches, compared with the original uncompressed image. (d) The compressed image as a single single patch with varying bond dimension ${\chi }_{\text{img}}=1,2,4,8$.

Figure 2

(a) The test accuracy for classification using MPS as a function of the image bond dimensions ${\chi }_{\text{img}}$ for fixed classifier bond dimension ${\chi }_{\text{class}}=10$. Each curve corresponds to a different number of patches; see Fig. 1. We report the average of the best 100 test accuracies to limit stochastic effects. The numbers in parentheses in the legend are the required number of qubits on a quantum computer. The dashed red line shows the single pixel limit, which is an exact encoding of the image with ${\chi }_{\text{img}}=1$. (b) The test accuracy as a function of the classifier bond dimension ${\chi }_{\text{class}}$, for fixed image bond dimension ${\chi }_{\text{img}}=4$.

Figure 3

(a) The quantum circuit encoding and classification scheme using 11 qubits. We compress each image using $M_{\text{img}}$ layers of sequential circuits consisting of two-qubit gates. The address and color qubits are indicated by circles, and a square, respectively, similar to Fig. 1. We classify each image with $M_{\text{class}}$ layers of sequential two-qubit gates (the case $M_{\text{img}}=M_{\text{img}}=2$ is shown for illustration). Before the final readout, three additional trainable two-qubit gates are added. (b) Compression of a selected image using $M_{\text{img}}=1,2,3$, compared with the uncompressed image.

Figure 4

The test accuracy using hardware-efficient quantum circuits for image compression and classifier on the Fashion-MNIST dataset. We again report the average of the best 100 converged iterations to limit stochastic effects. The accuracy is plotted against the number of layers $M_{\text{class}}$ in the quantum circuit classifier. We show the results the images compressed using $M_{\text{img}}=1,2,3$, compared with the exact uncompressed images. For reference, we show the accuracy we achieve for the ${\chi }_{\mathrm{img}}=4$ single patch MPS case with ${\chi }_{\mathrm{class}}=16$.

Figure 5

(a) Basic MPS diagrammatic notation. Connected tensors represents summing over the index corresponding to the connected legs. From left to right, a vector inner product, a matrix-vector multiplication, and a matrix-matrix multiplication. (b) Converting a general vector into an MPS. A generic vector with size $2^N$ can be viewed as an $N$ index tensor. We convert each vector into a product of matrices via a singular value decomposition, then truncate the singular value matrix (the red tensor) to the desired number of singular values (i.e., $\chi$). By repeatedly applying this procedure, we decompose the original vector into a matrix-product state with bond dimension $\chi$. To return to the vector representation, we would contract the MPS along the virtual indices ${\alpha }_j$ (albeit with some loss from the truncation).

Figure 6

(a) The mapping between an MPS tensor and a unitary matrix. The MPS tensors are used to define a unitary action on an additional set of qubits initialized to a product state, where the number of additional qubits depends on the bond dimension of the MPS. (b) An MPS with bond dimension $\chi$ is equivalent to a quantum circuit where each gate acts on $log\chi +1$ qubits. (c) Every $M$-layer sequential quantum circuit with two qubit gates is a subset of the set of sequential quantum circuit with $(M+1)$ qubit gates. Here, we show the equivalence for $M=3$.

Figure 7

(a) The training accuracy as a function of the image bond dimension. (b) The accuracy as a function of classifier bond dimension.

Figure 8

(a)–(c) Cross-entropy loss across training for the quantum circuit classifier. We show classifiers with $M_{\text{class}}=2,6$, and 12, respectively. In each case, we show $M_{\mathrm{img}}=1,2,$ and 3. (d) Training accuracy for the quantum circuit classifier. Each curve corresponds to a different $M_{\mathrm{img}}$.