We analyze the complexity of synthesizing random states and unitary operators in a multi-qudit system in two paradigms. In one case, we consider the situation in which we manipulate the system by applying a sequence of one- and two-qudit quantum gates that constitute the elementary, and universal, gate set. The minimum number of gates required to perform the desired operation represents the computational complexity. In the other case, we consider the situation in which we manipulate the physical system using physical fields with optimized control pulses. The minimum time required to perform the desired operation represents the physical complexity. In both cases, we use analytical arguments in combination with optimal-control-theory numerical calculations to determine the complexity of random operations. We show that the computational complexity of random states or unitary operators scales exponentially with the number of qudits. Our numerical results suggest that the physical complexity of preparing random quantum states and unitary operators scales more slowly than the computational complexity. We discuss various implications of our results, especially concerning the relationship between random and pseudorandom states and unitary operators.
We analyze the complexity of synthesizing random states and unitary operators in a multi-qudit system in two paradigms. In one case, we consider the situation in which we manipulate the system by applying a sequence of one- and two-qudit quantum gates that constitute the elementary, and universal, gate set. The minimum number of gates required to perform the desired operation represents the computational complexity. In the other case, we consider the situation in which we manipulate the physical system using physical fields with optimized control pulses. The minimum time required to perform the desired operation represents the physical complexity. In both cases, we use analytical arguments in combination with optimal-control-theory numerical calculations to determine the complexity of random operations. We show that the computational complexity of random states or unitary operators scales exponentially with the number of qudits. Our numerical results suggest that the physical complexity of preparing random quantum states and unitary operators scales more slowly than the computational complexity. We discuss various implications of our results, especially concerning the relationship between random and pseudorandom states and unitary operators.
We perform optimal-control-theory calculations to determine the minimum number of two-qubit controlled-not (cnot) gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that all produce the desired result, even at the minimum number of gates. We also show that the number of entangling gates can be reduced if we use multiqubit entangling gates instead of two-qubit cnot gates, as one might expect based on parameter counting calculations. In addition to treating the general case of arbitrary target states or unitary operators, we apply the numerical approach to the special case of synthesizing the multiqubit Toffoli gate. This approach can be used to investigate any other specific few-qubit task and provides insight into the tightness of different bounds in the literature.
Abstract The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.
Circuit synthesis of multi-controlled gates is crucial for qudit (d-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general (n−1)-controlled unitaries on n qudits, we reduce the controlled-increment (CINC) and generalized controlled-X (GCX) gate counts to O(n2), improving upon existing O(n2+log2d) CINC and O(n3) GCX bounds. For (n−1)-controlled special unitaries, this complexity is further reduced to O(n). Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from n to m qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when d is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general n-qudit unitaries.
Random many-body states and matrices are both a useful tool to model certain physical systems and an important asset for quantum computation. Realizing them, however, generally requires an exponential (in system size) amount of resources. Recent research has presented a way out by showing that one can generate random states and matrices, or more precisely a controlled approximation of them, by applying a quantum circuit built in terms of few-body unitary gates. Most of this research has been focussed on the case of quantum circuits composed by fully random unitary gates. Here we consider what happens for circuits that, instead, involve a minimal degree of randomness. Specifically, we concentrate on two different settings: (a) brickwork quantum circuits with a single one-qudit random matrix at a boundary; (b) brickwork quantum circuits with fixed interactions but random one-qudit gates everywhere. We show that, for any given initial state, (a) and (b) produce a distribution of states approaching the Haar distribution in the limit of large circuit depth. More precisely, we show that the moments of the distribution produced by our circuits can approximate the ones of the Haar distribution in a depth proportional to the system size. Interestingly we find that in both Cases (a) and (b) the relaxation to the Haar distribution occurs in two steps—this is in contrast with what happens in fully random circuits. Moreover, we show that choosing appropriately the fixed interactions, for example taking the local gate to be a dual-unitary gate with high enough entangling power, minimally random circuits produce a Haar random distribution more rapidly than fully random circuits. In particular, dual-unitary circuits with maximal entangling power—i.e. perfect tensors—appear to provide the optimal quantum state design preparation for any design number. Our results also suggest that the minimally random settings (a) and (b) give an optimal framework to prepare unitary designs, i.e., random unitary matrices.
We use a random search technique to find quantum gate sequences that implement perfect quantum state preparation or unitary operator synthesis with arbitrary targets. This approach is based on the recent discovery that there is a large multiplicity of quantum circuits that achieve unit fidelity in performing a given target operation, even at the minimum number of single-qubit and two-qubit gates needed to achieve unit fidelity. We show that the fraction of perfect-fidelity quantum circuits increases rapidly as soon as the circuit size exceeds the minimum circuit size required for achieving unit fidelity. This result implies that near-optimal quantum circuits for a variety of quantum information processing tasks can be identified relatively easily by trying only a few randomly chosen quantum circuits and optimizing their parameters. In addition to analyzing the case where the cnot gate is the elementary two-qubit gate, we consider the possibility of using alternative two-qubit gates. In particular, we analyze the case where the two-qubit gate is the B gate, which is known to reduce the minimum quantum circuit size for two-qubit operations. We apply the random search method to the problem of decomposing the four-qubit Toffoli gate and find a 15-cnot-gate decomposition.
The intersection of physics and information theory has advanced rapidly in the past few decades. Information-theoretic concepts have been employed to formulate physical laws, while quantum information processing, such as quantum computation and communication, has been developed both theoretically and experimentally. Simultaneously, understanding the universal or specific behaviors of quantum many-body systems has remained a central challenge since shortly after the inception of quantum mechanics. This is studied in part by statistical mechanics, which models complex many-body systems using simple ensembles. In this thesis, we investigate information-theoretic problems in quantum many-body dynamics that lie at the intersection of these avenues. To begin with, we investigate the extent of randomness, captured by unitary designs, in random quantum circuits. Quantum circuits serve as discrete models for random many-body quantum dynamics. Specifically, we study how the internal structure of random gates influences the generation of randomness in quantum circuits. We demonstrate that the spectral gap of structured random circuits can be larger than that of unstructured ones, implying that the former can converge to designs faster than the latter. In this analysis, free-fermionic techniques play a crucial role in obtaining exact results on the spectrum of second-moment operators. We then investigate the effect of quantum measurements on the quantum complexity in random circuits. Here, the quantum complexity of a state is defined as the minimum number of local gates required for its exact preparation in quantum circuits. We prove that in monitored random quantum circuits, the complexity undergoes a measurement-induced phase transition by leveraging techniques from percolation theory. Above the critical measurement rate, the complexity saturates quickly. Below the threshold, it grows at least linearly for an exponentially long time. Finally, we discuss quantum communication scenarios in noisy quantum dynamics, where we simulate the identity channel by combining a given noisy quantum dynamics and our quantum operational strategies. Specifically, we consider multi-time noisy processes and optimize our strategies to simulate near-noiseless channels, both of which are mathematically formulated using quantum combs. We introduce formal concepts of simulation and capacity for quantum combs, and we derive lower-bounds on the minimum simulation error in terms of semidefinite programming (SDP). We impose certain conditions on our strategies, such as the positive partial transpose (PPT) preserving constraint. We then analyze these SDPs for simple models, such as the depolarizing process, to determine their fundamental communication limits.