Advantages in several fields of research and industry are expected with the rise of quantum computers. However, the computational cost to load classical data in quantum computers can impose restrictions on possible quantum speedups. Known algorithms to create arbitrary quantum states require quantum circuits with depth O(N) to load an N-dimensional vector. Here, we show that it is possible to load an N-dimensional vector with exponential time advantage using a quantum circuit with polylogarithmic depth and entangled information in ancillary qubits. Results show that we can efficiently load data in quantum devices using a divide-and-conquer strategy to exchange computational time for space. We demonstrate a proof of concept on a real quantum device and present two applications for quantum machine learning. We expect that this new loading strategy allows the quantum speedup of tasks that require to load a significant volume of information to quantum devices.
Advantages in several fields of research and industry are expected with the rise of quantum computers. However, the computational cost to load classical data in quantum computers can impose restrictions on possible quantum speedups. Known algorithms to create arbitrary quantum states require quantum circuits with depth O(N) to load an N-dimensional vector. Here, we show that it is possible to load an N-dimensional vector with exponential time advantage using a quantum circuit with polylogarithmic depth and entangled information in ancillary qubits. Results show that we can efficiently load data in quantum devices using a divide-and-conquer strategy to exchange computational time for space. We demonstrate a proof of concept on a real quantum device and present two applications for quantum machine learning. We expect that this new loading strategy allows the quantum speedup of tasks that require to load a significant volume of information to quantum devices.
Among matrix decomposition methods, the two factorized matrices obtained by Non-negative matrix factorization (NMF) generally yield a more natural and interpretable representations. It is computationally expensive to find the two factorized matrices W and H with a fixed rank r from a non-negative matrix X∈Rn⇥m by minimizing ||X-WH^Τ ||^2. Even if the proposed separability assumption enables polynomial separable NMF (SNMF) algorithms, the computational cost is not affordable for the continuously growing big data. As a powerful technique to deal with high dimensional data, quantum computing possesses quantum advantages for solving machine learning problems, at which the quantum advantages include both an exponential speedup and a more efficient learning ability with respect to the classical counterparts. In this research, based on the divide-and-conquer anchoring (DCA) scheme, quantum DCA (QDCA) is devised to solve SNMF problems. Benefiting from the linear algebraic nature of quantum mechanics and the convenience that statistical distribution is natively available in quantum mechanics with a superset, QDCA achieves an exponential speedup with respect to the classical counterpart. The runtime complexity is O(poly log(n + m)), where n x m is the size of the input matrix. In addition, the proposed QDCA can naturally generalize to solve near-separable NMF under a polynomial logarithmic runtime. Specifically, the time consuming divide step can be completed by a quantum algorithm for linear operations with an exponential speedup. In the conquer step, considering that reading out quantum data into classical forms is exponential expensive and attributing to that anchors can be sampled with a high probability from a probability distribution in the resulting quantum states, the proposed heuristic post-selection method enables QDCA to extract information from quantum states efficiently, which also maintains the quantum speedup after measurements. Under a plausible assumption, QDCA performs efficiently, achieves the exponential speedup, and is beneficial for high dimensional problems. The QDCA is a hybrid quantum and classical algorithm that maintains the quantum speedup after measurements, which provides a new way to develop quantum machine learning algorithms.
A crucial subroutine in quantum computing is to load the classical data of N complex numbers into the amplitude of a superposed n=⌈log2N⌉-qubit state. It has been proven that any algorithm universally implementing this subroutine would need at least O(N) constant weight operations. However, the proof assumes that only n qubits are used, whereas the circuit depth could be reduced by extending the space and allowing ancillary qubits. Here we investigate this space-time tradeoff in quantum state preparation with classical data. We propose quantum algorithms with O(n2) circuit depth to encode any N complex numbers using only single-, two-qubit gates and local measurements with ancillary qubits. Different variances of the algorithm are proposed with different space and runtime. In particular, we present a scheme with O(N2) ancillary qubits, O(n2) circuit depth, and O(n2) average runtime, which exponentially improves the conventional bound. While the algorithm requires more ancillary qubits, it consists of quantum circuit blocks that only simultaneously act on a constant number of qubits and at most O(n) qubits are entangled. We also prove a fundamental lower bound $\mitΩ(n)$ for the minimum circuit depth and runtime with arbitrary number of ancillary qubits, aligning with our scheme with O(n2). The algorithms are expected to have wide applications in both near-term and universal quantum computing.
A crucial subroutine in quantum computing is to load the classical data of N complex numbers into the amplitude of a superposed $n=\ensuremath{\lceil}{log}_{2}N\ensuremath{\rceil}$-qubit state. It has been proven that any algorithm universally implementing this subroutine would need at least O(N) constant weight operations. However, the proof assumes that only n qubits are used, whereas the circuit depth could be reduced by extending the space and allowing ancillary qubits. Here we investigate this space-time tradeoff in quantum state preparation with classical data. We propose quantum algorithms with O(n2) circuit depth to encode any N complex numbers using only single- and two-qubit gates, and local measurements with ancillary qubits. Different variances of the algorithm are proposed with different space and runtime. In particular, we present a scheme with O(N2) ancillary qubits, O(n2) circuit depth, and O(n2) average runtime, which exponentially improves the conventional bound. While the algorithm requires more ancillary qubits, it consists of quantum circuit blocks that only simultaneously act on a constant number of qubits, and at most O(n) qubits are entangled. We also prove a fundamental lower bound $\mathrm{\ensuremath{\Omega}}(n)$ for the minimum circuit depth and runtime with an arbitrary number of ancillary qubits, aligning with our scheme with O(n2). The algorithms are expected to have wide applications in both near-term and universal quantum computing.
Quantum state preparation is an important subroutine for quantum computing. We show that any n-qubit quantum state can be prepared with a Θ(n)-depth circuit using only single- and two-qubit gates, although with a cost of an exponential amount of ancillary qubits. On the other hand, for sparse quantum states with d⩾2 nonzero entries, we can reduce the circuit depth to Θ(log(nd)) with O(ndlogd) ancillary qubits. The algorithm for sparse states is exponentially faster than best-known results and the number of ancillary qubits is nearly optimal and only increases polynomially with the system size. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories, and find cases with exponential reductions of the circuit depth for all these three tasks. In particular, using our algorithm, we find a family of linear system solving problems enjoying exponential speedups, even compared to the best-known quantum and classical dequantization algorithms.
Broadly applicable quantum advantage, particularly in classical data processing and machine learning, has been a fundamental open problem. In this work, we prove that a small quantum computer of polylogarithmic size can perform large-scale classification and dimension reduction on massive classical data by processing samples on the fly, whereas any classical machine achieving the same prediction performance requires exponentially larger size. Furthermore, classical machines that are exponentially larger yet below the required size need superpolynomially more samples and time. We provide evidence for these quantum advantages in real-world applications, including single-cell RNA sequencing and movie review sentiment analysis, demonstrating four to six orders of magnitude reduction in size with fewer than 60 logical qubits. These quantum advantages are enabled by quantum oracle sketching, an algorithm for accessing the classical world in quantum superposition using only random classical data samples. Combined with classical shadows, our algorithm circumvents the data loading and readout bottleneck to construct succinct classical models from massive classical data, a task provably impossible for any classical machine that is not exponentially larger than the quantum machine. These quantum advantages persist even when classical machines are granted unlimited time or if BPP = BQP, and rely only on the correctness of quantum mechanics. Together, our results establish machine learning on classical data as a broad and natural domain of quantum advantage and a fundamental test of quantum mechanics at the complexity frontier.
Abstract Quantum machine learning (QML) is an emerging field that investigates the capabilities of quantum computers for learning tasks. While QML models can theoretically offer advantages such as exponential speed-ups, challenges in data loading and the ability to scale to relevant problem sizes have prevented demonstrations of such advantages on practical problems. In particular, the encoding of arbitrary classical data into quantum states usually comes at a high computational cost, either in terms of qubits or gate count. However, real-world data typically exhibits some inherent structure (such as image data) which can be leveraged to load them with a much smaller cost on a quantum computer. This work further develops an efficient algorithm for finding low-depth quantum circuits to load classical image data as quantum states. To evaluate its effectiveness, we conduct systematic studies on the MNIST, Fashion-MNIST, CIFAR-10, and Imagenette datasets. The corresponding circuits for loading the full large-scale datasets are available publicly as PennyLane datasets and can be used by the community for their own benchmarks. We further analyze the performance of various quantum classifiers, such as quantum kernel methods, parameterized quantum circuits, and tensor-network classifiers, and we compare them to convolutional neural networks. In particular, we focus on the performance of the quantum classifiers as we introduce nonlinear functions of the input state, e.g. by letting the circuit parameters depend on the input state.