We present two new algorithms that universally construct a quantum circuit approximating a controlled-U(t) up to a global phase factor of U(t), denoted by ΛθU(t), where U(t) = e−iHt is the unitary evolution operator of an unknown Hamiltonian H of the system, and is given as a black box with tunable parameter t. We show that it is possible to approximately construct ΛθU(t) with arbitrarily high accuracy. This algorithm is based on the dynamical decoupling effect [1]. If another black box consisting of a time-inverted version of U(t), namely U(−t), is also available we can implement ΛθU(t) exactly with arbitrarily high success probability. This algorithm is based on the quantum Zeno effect.
We present two new algorithms that universally construct a quantum circuit approximating a controlled-U(t) up to a global phase factor of U(t), denoted by ΛθU(t), where U(t) = e−iHt is the unitary evolution operator of an unknown Hamiltonian H of the system, and is given as a black box with tunable parameter t. We show that it is possible to approximately construct ΛθU(t) with arbitrarily high accuracy. This algorithm is based on the dynamical decoupling effect [1]. If another black box consisting of a time-inverted version of U(t), namely U(−t), is also available we can implement ΛθU(t) exactly with arbitrarily high success probability. This algorithm is based on the quantum Zeno effect.
Unlike fixed designs, programmable circuit designs support an infinite number of operators. The functionality of a programmable circuit can be altered by simply changing the angle values of the rotation gates in the circuit. Here, we present a new quantum circuit design technique resulting in two general programmable circuit schemes. The circuit schemes can be used to simulate any given operator by setting the angle values in the circuit. This provides a fixed circuit design whose angles are determined from the elements of the given matrix-which can be non-unitary-in an efficient way. We also give both the classical and quantum complexity analysis for these circuits and show that the circuits require a few classical computations. For the electronic structure simulation on a quantum computer, one has to perform the following steps: prepare the initial wave function of the system; present the evolution operator U = e(-iHt) for a given atomic and molecular Hamiltonian H in terms of quantum gates array and apply the phase estimation algorithm to find the energy eigenvalues. Thus, in the circuit model of quantum computing for quantum chemistry, a crucial step is presenting the evolution operator for the atomic and molecular Hamiltonians in terms of quantum gate arrays. Since the presented circuit designs are independent from the matrix decomposition techniques and the global optimization processes used to find quantum circuits for a given operator, high accuracy simulations can be done for the unitary propagators of molecular Hamiltonians on quantum computers. As an example, we show how to build the circuit design for the hydrogen molecule.
Due to increasing demands on speed and security of data processing, along with requirements on measurement precision in fundamental research, quantum phenomena are expected to play an increasing role in future technologies. Special attention must hence be paid to omnipresent decoherence effects, which hamper quantumness. Their consequence is always a deviation of the quantum state evolution (error) with respect to the expected unitary evolution if these effects are absent. In operational tasks such as the preparation, transformation, transmission, and detection of quantum states, these effects are detrimental and must be suppressed by strategies known as dynamical decoupling, or the more general dynamical control by modulation developed by us. The underlying dynamics must be Zeno-like, yielding suppressed coupling to the bath. There are, however, tasks which cannot be implemented by unitary evolution, in particular those involving a change of the system’s state entropy. Such tasks necessitate efficient coupling to a bath for their implementation. Examples include the use of measurements to cool (purify) a system, to equilibrate it, or to harvest and convert energy from the environment. If the underlying dynamics is anti-Zeno like, enhancement of this coupling to the bath will occur and thereby facilitate the task, as discovered by us. A general task may also require state and energy transfer, or entanglement of noninteracting parties via shared modes of the bath which call for maximizing the shared (two-partite) couplings with the bath, but suppressing the single-partite couplings. For such tasks, a more subtle interplay of Zeno and anti-Zeno dynamics may be optimal. We have therefore constructed a general framework for optimizing the way a system interacts with its environment to achieve a desired task. This optimization consists in adjusting a given “score” that quantifies the success of the task, such as the targeted fidelity, purity, entropy, entanglement, or energy by dynamical modification of the system-bath coupling spectrum on demand.
This paper explores a novel approach to quantum computation centered around self-referential quantum circuits. The core concept involves designing circuits that recursively incorporate themselves into their operational structure, effectively creating computational modules capable of self-replication. This is achieved through the strategic implementation of feedback loops and entanglement within the circuit architecture. The potential benefits of this paradigm include a reduction in circuit complexity and the possibility of constructing universal quantum computers from fundamentally simpler components. The theoretical framework presented herein outlines the key principles and mathematical underpinnings of this self-referential design, proposing a pathway toward more efficient and scalable quantum computation. The presented model uses the following notation: |ψ⟩ represents a quantum state, U(θ) represents a unitary operator with rotation angle θ, C represents a quantum circuit, and E represents the expectation value. Specifically, the expectation value of an observable A is denoted as E[A] = <ψ|A|ψ>, where |ψ> is the quantum state. The Hamiltonian of a quantum system is typically represented as H, and the time evolution of the state is governed by the Schrödinger equation: iℏ∂|ψ(t)⟩/∂t = H|ψ(t)⟩. Entanglement is represented by the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. Quantum gates, such as the Hadamard gate (H) and the CNOT gate (CN), are represented as unitary transformations. The circuit representation involves sequential application of gates, and the overall unitary transformation is the product of these gates. The circuit depth, denoted as d, represents the number of consecutive gate operations applied to a quantum state. The circuit complexity, denoted as C, is a measure of the number of gates required to implement a specific quantum algorithm. The circuit fidelity, denoted as F, quantifies the accuracy of a quantum circuit's output, ranging from 0 to 1. We define the quantum state preparation as |ψ⟩ = ∑i ci|i⟩, where ci are complex coefficients and |i⟩ are the computational basis states. The unitary transformation is defined as U|ψ⟩ = ∑i (Uici)|i⟩, where Ui are the matrix elements of the unitary operator U. The propagation of quantum states through a circuit is represented by the Schrödinger equation, and the final state is obtained by applying the unitary transformation to the initial state. The concept of quantum coherence, denoted as χ, represents the ability of a quantum system to maintain its superposition state. The fidelity of a quantum state is defined as the probability that a measurement will yield the expected outcome.
Given a quantum gate implementing a d-dimensional unitary operation U_{d}, without any specific description but d, and permitted to use k times, we present a universal probabilistic heralded quantum circuit that implements the exact inverse U_{d}^{-1}, whose failure probability decays exponentially in k. The protocol employs an adaptive strategy, proven necessary for the exponential performance. It requires that k≥d-1, proven necessary for the exact implementation of U_{d}^{-1} with quantum circuits. Moreover, even when quantum circuits with indefinite causal order are allowed, k≥d-1 uses are required. We then present a finite set of linear and positive semidefinite constraints characterizing universal unitary inversion protocols and formulate a convex optimization problem whose solution is the maximum success probability for given k and d. The optimal values are computed using semidefinite programing solvers for k≤3 when d=2 and k≤2 for d=3. With this numerical approach we show for the first time that indefinite causal order circuits provide an advantage over causally ordered ones in a task involving multiple uses of the same unitary operation.
Dephasing--phase randomization of a quantum superposition state--is a major obstacle for the realization of high fidelity quantum logic operations. Here, we implement a two-qubit controlled-NOT gate using dynamical decoupling (DD), despite the gate time being more than 1 order of magnitude longer than the intrinsic coherence time of the system. For realizing this universal conditional quantum gate, we have devised a concatenated DD sequence that ensures robustness against imperfections of DD pulses that otherwise may destroy quantum information or interfere with gate dynamics. We compare its performance with three other types of DD sequences. These experiments are carried out using a well-controlled prototype quantum system--trapped atomic ions coupled by an effective spin-spin interaction. The scheme for protecting conditional quantum gates demonstrated here is applicable to other physical systems, such as nitrogen vacancy centers, solid state nuclear magnetic resonance, and circuit quantum electrodynamics.