Parameterized quantum circuits are the core of new technologies such as variational quantum algorithms and quantum machine learning, which makes studying its properties a valuable task. We implement parameterized circuits with definite and indefinite causal order and compare their performance under particular descriptors. One of these is the expressibility, which measures how uniformly a given quantum circuit can reach the whole Hilbert space. Another property that we focus on this work is the entanglement capability, more specifically the concurrence and the entangling power. We also find the causal relation between the qubits of our system with the quantum average causal effect measure. We have found that indefinite circuits offer expressibility advantages over definite ones while maintaining the level of entanglement generation. Our results also point to the existence of a correlation between the quantum average causal effect and the entangling power.
Parameterized quantum circuits are the core of new technologies such as variational quantum algorithms and quantum machine learning, which makes studying its properties a valuable task. We implement parameterized circuits with definite and indefinite causal order and compare their performance under particular descriptors. One of these is the expressibility, which measures how uniformly a given quantum circuit can reach the whole Hilbert space. Another property that we focus on this work is the entanglement capability, more specifically the concurrence and the entangling power. We also find the causal relation between the qubits of our system with the quantum average causal effect measure. We have found that indefinite circuits offer expressibility advantages over definite ones while maintaining the level of entanglement generation. Our results also point to the existence of a correlation between the quantum average causal effect and the entangling power.
Abstract An active area of investigation in the search for quantum advantage is quantum machine learning. Quantum machine learning, and parameterized quantum circuits in a hybrid quantum-classical setup in particular, could bring advancements in accuracy by utilizing the high dimensionality of the Hilbert space as feature space. But is the ability of a quantum circuit to uniformly address the Hilbert space a good indicator of classification accuracy? In our work, we use methods and quantifications from prior art to perform a numerical study in order to evaluate the level of correlation. We find a moderate to strong correlation between the ability of the circuit to uniformly address the Hilbert space and the achieved classification accuracy for circuits that entail a single embedding layer followed by 1 or 2 circuit designs. This is based on our study encompassing 19 circuits in both 1- and 2-layer configurations, evaluated on 9 datasets of increasing difficulty. We also evaluate the correlation between entangling capability and classification accuracy in a similar setup, and find a weak correlation. Future work will focus on evaluating if this holds for different circuit designs.
In a conventional circuit for quantum machine learning, the quantum gates used to encode the input parameters and the variational parameters are constructed with a fixed order. The resulting output function, which can be expressed in the form of a restricted Fourier series, has limited flexibility in the distributions of its Fourier coefficients. This indicates that a fixed order of quantum gates can limit the performance of quantum machine learning. Building on this key insight (also elaborated with examples), we introduce indefinite causal order to quantum machine learning. Because the indefinite causal order of quantum gates allows for the superposition of different orders, the performance of quantum machine learning can be significantly enhanced. Considering that the current accessible quantum platforms only allow one to simulate a learning structure with a fixed order of quantum gates, we reform the existing simulation protocol to implement indefinite causal order and further demonstrate the positive impact of indefinite causal order on specific learning tasks. Our results offer useful insights into possible quantum effects in quantum machine learning.
Abstract Parameterized quantum circuits (PQCs) play an essential role in the performance of many variational quantum algorithms. One challenge in implementing such algorithms is choosing an effective circuit that well represents the solution space while maintaining a low circuit depth and parameter count. To characterize and identify expressible, yet compact, circuits, several descriptors are proposed, including expressibility and entangling capability, that are statistically estimated from classical simulations. These descriptors are computed for different circuit structures, varying the qubit connectivity and selection of gates. From these simulations, circuit fragments that perform well with respect to the descriptors are identified. In particular, a substantial improvement in performance of two‐qubit gates in a ring or all‐to‐all connected arrangement, compared to that of those on a line, is observed. Furthermore, improvement in both descriptors is achieved by sequences of controlled X‐rotation gates compared to sequences of controlled Z‐rotation gates. In addition, it is investigated how expressibility “saturates” with increased circuit depth, finding that the rate and saturated value appear to be distinguishing features of a PQC. While the correlation between each descriptor and algorithm performance remains to be investigated, methods and results from this study can be useful for algorithm development and design of experiments.
Variational quantum algorithms are one of the promising methods to obtain quantum advantage in the noisy intermediate scale quantum computers era.They rely on a classical optimization procedure, a cost function and a parameterized quantum circuit to build the solution of a particular problem.Most of the work regarding the quantum circuits part is based on heuristic propositions for the circuit structure and reside only within the borders of VQA applications.In this context, the main objective of our work was the characterization of entanglement generation and distribution of generated states for different PQCs structures.Applying the mean entanglement considering the Scott entanglement measures and the expressibility quantifier, we studied the behavior of 5 possible connectivities between qubits that appear in the contemporary quantum computers: No connections, linear, ring, star and all-to-all, for different number of qubits and circuit concatenations (layers).For two circuit architectures with different local parameterizations, we discussed how entanglement and expressibility are connected, showing that the entanglement generation for only 1 layer is influential for the expressibility evolution as a function of the number of layers.Circuits generating mean and standard deviation for entanglement closer to the uniformly distributed states at 1 layer will have a steeper evolution of expressibility.This result is affected by the local parameterization and number of qubits.We then compared the circuits generated entanglement with the entanglement of paradigmatic states EP R n , GHZ n and W n to understand the entanglement characteristics of the different connections.The results showed how the different connectivities will influence parameterized quantum circuits for applications in VQAs and also that these can present the behavior of pseudorandom quantum circuits.
Abstract While scalable error correction schemes and fault tolerant quantum computing seem not to be universally accessible in the near sight, the efforts of many researchers have been directed to the exploration of the contemporary available quantum hardware. Due to these limitations, the depth and dimension of the possible quantum circuits are restricted. This motivates the study of circuits with parameterized operations that can be classically optimized in hybrid methods as variational quantum algorithms, enabling the reduction of circuit depth and size. The characteristics of these Parameterized Quantum Circuits (PQCs) are still not fully understood outside the scope of their principal application, motivating the study of their intrinsic properties. In this work, we analyse the generation of random states in PQCs under restrictions on the qubits connectivities, justified by different quantum computer architectures. We apply the expressibility quantifier and the average entanglement as diagnostics for the characteristics of the generated states and classify the circuits depending on the topology of the quantum computer where they can be implemented. As a function of the number of layers and qubits, circuits following a Ring topology will have the highest entanglement and expressibility values, followed by Linear/All-to-all almost together and the Star topology. In addition to the characterization of the differences between the entanglement and expressibility of these circuits, we also place a connection between how steep is the increase on the uniformity of the distribution of the generated states and the generation of entanglement. Circuits generating average and standard deviation for entanglement closer to values obtained with the truly uniformly random ensemble of unitaries present a steeper evolution when compared to others.
Variational quantum algorithms are one of the promising methods to obtain quantum advantage in the noisy intermediate scale quantum computers era. They rely on a classical optimization procedure, a cost function and a parameterized quantum circuit to build the solution of a particular problem. Most of the work regarding the quantum circuits part is based on heuristic propositions for the circuit structure and reside only within the borders of VQA applications. In this context, the main objective of our work was the characterization of entanglement generation and distribution of generated states for different PQCs structures. Applying the mean entanglement considering the Scott entanglement measures and the expressibility quantifier, we studied the behavior of 5 possible connectivities between qubits that appear in the contemporary quantum computers: No connections, linear, ring, star and all-to-all, for different number of qubits and circuit concatenations (layers). For two circuit architectures with different local parameterizations, we discussed how entanglement and expressibility are connected, showing that the entanglement generation for only 1 layer is influential for the expressibility evolution as a function of the number of layers. Circuits generating mean and standard deviation for entanglement closer to the uniformly distributed states at 1 layer will have a steeper evolution of expressibility. This result is affected by the local parameterization and number of qubits. We then compared the circuits generated entanglement with the entanglement of paradigmatic states EPRn, GHZn and Wn to understand the entanglement characteristics of the different connections. The results showed how the different connectivities will influence parameterized quantum circuits for applications in VQAs and also that these can present the behavior of pseudorandom quantum circuits.