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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.

where \(\vert {\psi _j}\rangle \) are unit vectors. We propose an algorithm to load an N-dimensional vector in a quantum state as shown in Eq. (2) using a circuit with \(O(\log _2^2(N))\) depth and O(N) qubits. The devised method is based on quantum forking13,14 and uses a divide-and-conquer strategy15. The circuit depth is decreased at the cost of increasing the circuit width and creating entanglement between data register qubits and an ancillary system. Thus when the data register is considered alone (i.e. by tracing out the ancilla qubits), the resulting state is mixed and not equal to the pure state shown in Eq. (1). However, it is important to note that in Eq. (2) the classical data is still encoded as probability amplitudes of an orthonormal basis set. Useful applications can be constructed based on this, and we provide two example applications in machine learning and statistical analysis.

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The divide-and-conquer paradigm is used in efficient algorithms for sorting16, computing the discrete Fourier transform17, and others15. The main idea is to divide a problem into subproblems of the same class and combine the solutions of the subproblems to obtain the solution of the original problem. The circuit based divide-and-conquer state preparation algorithm has computational cost O(N) and the total complexity time is \(O_c(N) + O_q(\log _2^2(N))\), where \(O_c(N)\) is classical pre-computation time to create the quantum circuit that will load the information in the quantum device and \(O_q(\log _2^2(N))\) is the depth of the quantum circuit. With the supposition that we will load the input vector \(m\gg N\) times, the amortized computational time to load the real vector is \(O_q(\log _2^2(N))\). The modified version of the loading problem allows an exponential advantage in the depth of the quantum circuit using O(N) qubits.

The construction of the quantum state in the previous section starts in the root of state-tree \(\vert {0}\rangle _n\) and build the states in each level of the state-tree in a top-down strategy until to build the state described by the last level of the state-tree. In this Section, we propose a divide-and-conquer load strategy, and the desired quantum state is built following a bottom-up strategy. First, we divide the input into bidimensional subvectors and load qubits corresponding to the normalized bidimensional subvectors. In the next steps, we generate the subvectors of the previous levels.

Algorithm 3 presents the complete pseudocode for the divide-and-conquer state preparation algorithm. The for loop in line 2 initializes the qubit q[k] with the value \(R_y(\alpha _k)\). After this step, the qubits with index \(k > \lfloor (N-1)/2\rfloor \) (in the leaf of the angle tree) are normalized versions of the states in the leafs of the state-tree. The next subroutine with \(R_z\) rotations (Line 4 to Line 5) is used to encode phase information. Line 6 calculates the index of the first angle that has a right children in the angle-tree data structure. The while loop starting at line 7 combines the states generated in the subtree rooted in the angle \(\alpha _{actual}\). To combine the states, we first apply a cswap(q[actual], q[left_child], q[right_child]), and then we update the values of left and right child with the value of their left child and apply another cswap(q[actual], q[left_child], q[right_child]) while the left_child and right_child have valid values. With the input described by the angle-tree in Fig. 1a, Algorithm 3 generates the circuit described in Fig. 4.

Rotated angle-tree and a circuit generated by the divide-and-conquer strategy described in Algorithm 3. The quantum bit q[k] in the circuit is aligned with the angle \(\alpha [k]\) in the angle-tree, this organization allows to draw the quantum gates in each layer in parallel. In this example, the desired state is stored in qubits q[0], q[1] and q[3] to generate the quantum state with entangles ancilla as in Eq. (2).

The ancillary states \(\vert {\psi _0}\rangle , \dots , \vert {\psi _{N-1}}\rangle \) in Eq. (2) are not necessarily orthogonal to each other, but we can modify the divide-and-conquer state preparation adding label qubits to ensure orthonormality of the ancillary states with the addition of label quantum register with \(\log _2(N)\) qubits. The label register is prepared in \(|0\rangle ^{\otimes \log _2(N)}\), and \(\log _2(N)\) controlled-NOT (CNOT) gates are applied to the label qubits, each controlled by a data qubit. With this modification, the final state becomes

The main difference between the divide-and-conquer state preparation and previous approaches is an exchange between circuit depth by circuit width. Table 1 presents the depth of the circuits generated using the proposed strategy, implementation of a version of11 available at19 and a non optimized version of the algorithm described in10. The proposed strategy and10 implementation are publicly available. The implementation of the proposed method shows its theoretical asymptotic time advantage to load a vector when the dimension is larger than 32. The proposed method has two main disadvantages: the linear number of qubits in relation to the logarithmic number in other methods, and the information entangled in the ancillary qubits.

The higher depth of circuits using the divide-and-conquer strategy with small vectors occurs because of the use of three-qubits gates to combine the vectors. In other works, it is only necessary to use O(n) qubits to load a \(2^n\)-dimensional vector while requiring sequential applications of \(O(2^n)\) n-controlled gates. To improve the performance of the divide-and-conquer loading strategy and to reduce the number of qubits one can combine algorithm11 with the divide-and-conquer strategy. Instead of divide the vector in parts with size 2, we can divide the vector in parts with size k (equal to a power of 2), load the normalized k-dimensional vectors using a sequential algorithm and combine the small vectors with the divide-and-conquer approach.

This section compares the divide-and-conquer algorithm with two other approaches in which input data encoding in a quantum state can be achieved to initialize a quantum circuit, namely qubit encoding and amplitude encoding. In the former, data is encoded in the amplitudes of individual qubits in a fully separable state, performed using single-qubit rotations20. In the later, data is encoded in the amplitudes of an entangled state11,18, similarly to the divide-and-conquer. We use the accuracy of a quantum variational classifier as a metric to evaluate the state preparations. The divide-and-conquer algorithm is expected to produce results similar to the amplitude encoding. The results of the classifier using qubit encoding are also presented for completeness, albeit our main objective is to compare the divide-and-conquer and amplitude encoding schemes.

The results show similar classification accuracy for all encodings, favoring qubit encoding due to the greater number of circuit parameters for the optimization. The main advantage of divide-and-conquer encoding over qubit encoding is the representation of encoded data in a quantum state of a reduced number of qubits, \(\log _2 (N)\), compared to the initial state \(N-1\). This also results in a lower depth classifier. Moreover, when the data is given by qubit encoding, TTN circuits can be evaluated efficiently using classical techniques20. This is not true when the input data is amplitude encoded. The advantage over amplitude encoding is a lower depth encoding circuit for \(N\ge 64\) (Table 1).

To verify that the above comparison of the models is appropriate, a nonparametric statistical test was employed. We used the Wilcoxon paired signed-rank test25 with \(\alpha =0.05\) to check whether there exist significant differences between the classification performances of compared encoders over the chosen datasets. As expected, we verified that amplitude encoding and divide-and-conquer encoding are statistically equivalent for all datasets.

One of the major open problems for practical applications of quantum computing is to develop an efficient means to encode classical data in a quantum state3. Most quantum algorithms do not present advantages in loading data2. The method proposed in this work fills this gap by proposing a new quantum state preparation paradigm, which can complement or enhance the known methods, such as qubit encoding and amplitude encoding. Our approach was based on the Mttnen et al. algorithm10 and a divide-and-conquer approach using controlled swap gates and ancilla qubits. With this modification, we obtain an exponential quantum speedup in time to load a N-dimensional real vector in the amplitude of a quantum state with a quantum circuit of depth \(O(\log _2^2(N))\) and space O(N). The exponential speedup to load data in quantum devices has a potential impact on speeding up the solution of problems in quantum machine learning and other quantum algorithms that need to load data from classical devices. ff782bc1db