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Removal of barren plateaus in QCBMs inside our synergistic optimization framework as demonstrated by the gradient variances with respect to the KL divergence loss in Eq. (3). The grey lines indicate the gradient variances of linear topology circuits, whereas the blue lines indicate gradient variances of all-to-all topology circuits. The numbers at the beginning or the end of the lines denote the number of qubits. We record the median gradient variances over 1000 repetitions for the randomly initialized circuits (left), and 100 repetitions for the χ = 2 MPS initialized circuits (right), as well as bootstrapped 25-75 percentile confidence intervals of the median inside the shaded areas. We study the Cardinality N/2 dataset for the respective number of qubits N . The gradients are measured with respect to the YY-entangling gate contribution of the first SU(4) gate in the circuit between qubit 1 and 2. For random parameter initializations, the gradient variance decays exponentially in the number of qubits, and also the circuit depth up until a certain limit. This is clear indication for the existence of barren plateaus. One all-to-all layer of SU(4) gates appears to fully maximize the degree of barrenness. In contrast, the gradient variances of MPS initialized circuits neither decay significantly in the number of qubits nor with increasing circuit depth.
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While recent breakthroughs have proven the ability of noisy intermediate-scale quantum (NISQ) devices to achieve quantum advantage in classically-intractable sampling tasks, the use of these devices for solving more practically relevant computational problems remains a challenge. Proposals for attaining practical quantum advantage typically involve...
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... assess whether the synergistic framework is expected to be effective at improving the trainability of PQCs as the number of qubits increases, we now assess the variance of parameter gradients, i.e. the barrenness, of QCBMs training on the cardinality dataset. The results are shown in Fig. 3. We probe the gradient of the KL divergence loss with respect to the parameter controlling the YY-entangling component (according to the KAK-decomposition [62]) of the first SU(4) gate between qubits 1 and 2 (see Appendix C 2 for details). Gradient magnitudes for that parameter are recorded 1000 times per data point in the case of ...
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... removal of barren plateaus, as indicated in Fig. 3, is vital to ensuring the trainability of PQCs and their viability on quantum hardware. Vanishing gradient variances imply that gradient magnitudes also vanish [19], which leads the estimation of parameter gradients on quantum hardware to require a number of measurements which grows exponentially in the number of qubits. Additionally, ...
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... hinder the ability of gradient-based and gradient-free optimizers to find high-quality solutions, as well as the existence of large numbers of low-quality local minima [24], which present further difficulties in learning. Aside from improving the training performance in practice (as seen in Fig. 2), stable gradient variances (such as those in Fig. 3) hint that a finite (or at worst, non-exponential) number of quantum circuit evaluations may be sufficient to estimate parameter gradients and perform PQC optimization on quantum hardware in a scalable ...
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... to the quantum circuit. Interestingly, with random initial parameters, the linearly arranged circuit reaches better KL divergence values (black, left panel) than when the final layer is extended to an all-to-all topology (black, right panel). This highlights the deterioration of trainability with increasing circuit depth that can also seen in Fig. 3. The near-identity initialization is notably less hampered by this. When classically initializing the QCBMs, the models can only aim to recover the MPS performance when the circuit is not extended with additional gates FIG. 4. Optimization results for QCBMs training on the Cardinality dataset with N = 10 qubits and three linear layers. ...
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... changes in parametrization. Therefore, we chose σ cma = 10 −2 for the random and near-identity initializations, and σ cma = 7.5·10 −3 , σ cma = 5.0·10 −3 , and σ cma = 2.5·10 −3 for the MPS initialized modes with χ = 2, χ = 4, and χ = 8, respectively. The population sizes λ cma were always chosen to be λ cma = 20, meaning that each iteration in Fig. 3 corresponds to 20 quantum circuit simulations. In addition to a limit to the number of optimization steps, i.e., 10000, 15000, and 15000 for the PQCs in Fig. 2 (for Cardinality, BAS, and the Heisenberg model respectively), we also set a loss tolerance of 5·10 −4 which may stop the optimization if differences of loss values between ...
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... Achieving such quantum-inspired advantage would be a promising step towards demonstrating practical quantum advantage with near-term devices. This could be possible by exploiting recent efficient proposals [71,72] to map TNs to quantum circuit generative models such as quantum-circuit Born machines [73] . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t For our purposes, we shall be considering the MPS with open boundary conditions (OBC) [75] so that the left-and rightmost tensors are actually vectors so that χ 0 = χ N = 1 (so that the trace is not needed). A perspective that is useful when dealing with symmetry group transformations on tensors, is to view tensors as linear maps from an input vector space V in to output vector space V out . ...
Constrained combinatorial optimization problems abound in industry, from portfolio optimization to logistics. One of the major roadblocks in solving these problems is the presence of non-trivial hard constraints which limit the valid search space. In some heuristic solvers, these are typically addressed by introducing certain Lagrange multipliers in the cost function, by relaxing them in some way, or worse yet, by generating many samples and only keeping valid ones, which leads to very expensive and inefficient searches. In this work, we encode arbitrary integer-valued equality constraints of the form A⃗x = ⃗b, directly into U (1) symmetric tensor networks and leverage their applicability as quantum-inspired generative models to assist in the search of solutions to combinatorial optimization problems within the generator-enhanced optimization (GEO) framework of Ref. [1]. This allows us to exploit the generalization capabilities of TN generative models while constraining them so that they only output valid samples. Our constrained TN generative model efficiently captures the constraints by reducing number of parameters and computational costs. We find that at tasks with constraints given by arbitrary equalities, symmetric Matrix Product States outperform their standard unconstrained counterparts at finding novel and better solutions to combinatorial optimization problems.
The classification of Electroencephalogram (EEG) signals into distinct frequency bands is a critical task in understanding brain function and diagnosing neurological disorders. The information obtained from frequency-specific classification has multiple applications, such as frequency-based wheelchair control, frequency-based 36-stroke brain operated keyboard for paralysed patients etc. In this work, a method based on machine learning to develop the frequency-based classification of EEG signals is proposed. The performance of Classical Machine Learning (CML) algorithms and Quantum Machine Learning (QML) techniques for the classification of EEG signals across four frequency bands are investigated. The primary objective is to evaluate the performance of QML models against traditional CML models in terms of computational efficiency, time efficiency and accuracy and uncover potential benefits offered by quantum computing for a particular task of classifying EEG signals. The goal is to assess the advantages of using quantum algorithms for classifying EEG signals. This includes improving accuracy and enhancing efficiency. These findings add to the existing knowledge about how quantum machine learning can benefit neuroscience in terms of enhancing methods that rely on EEG data.
Tensor networks (TNs) are a family of computational methods built on graph-structured factorizations of large tensors, which have long represented state-of-the-art methods for the approximate simulation of complex quantum systems on classical computers. The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed TN algorithms to scale to even larger quantum simulation problems, and to be employed more broadly for solving machine learning tasks. The ``quantum-inspired'' nature of TNs permits them to be mapped to parametrized quantum circuits (PQCs), a fact which has inspired recent proposals for enhancing the performance of TN algorithms using near-term quantum devices, as well as enabling joint quantum-classical training frameworks that benefit from the distinct strengths of TN and PQC models. However, the success of any such methods depends on efficient and accurate methods for approximating TN states using realistic quantum circuits, which remains an unresolved question.
This work compares a range of novel and previously-developed algorithmic protocols for decomposing matrix product states (MPS) of arbitrary bond dimension into low-depth quantum circuits consisting of stacked linear layers of two-qubit unitaries. These protocols are formed from different combinations of a preexisting analytical decomposition method together with constrained optimization of circuit unitaries, with initialization by the former method helping to avoid poor-quality local minima in the latter optimization process. While all of these protocols have efficient classical runtimes, our experimental results reveal one particular protocol employing sequential growth and optimization of the quantum circuit to outperform all others, with even greater benefits in the setting of limited computational resources. Given these promising results, we expect our proposed decomposition protocol to form a useful ingredient within any joint application of TNs and PQCs, further unlocking the rich and complementary benefits of classical and quantum computation.
