scholarly journals Measuring the Tangle of Three-Qubit States

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 436 ◽  
Author(s):  
Adrián Pérez-Salinas ◽  
Diego García-Martín ◽  
Carlos Bravo-Prieto ◽  
José Latorre
Keyword(s):  
Direct Measurement ◽  
Canonical Form ◽  
Quantum Circuit ◽  
Quantum Gates ◽  
Tripartite Entanglement ◽  
Single Qubit ◽  
Optimal Values ◽  
Random States ◽  
Qubit States ◽  
Computational Basis

We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.

Experimental demonstration of optimized quantum process tomography on the IBM quantum experience

2020 ◽  
pp. 2040004
Author(s):  
Akshay Gaikwad ◽  
Krishna Shende ◽  
Kavita Dorai
Keyword(s):  
Quantum Circuit ◽  
Quantum Gates ◽  
Experimental Demonstration ◽  
Superconducting Qubit ◽  
Quantum Process ◽  
Single Qubit ◽  
Quantum Process Tomography ◽  
Process Tomography ◽  
Quantum Processor ◽  
Least Squares Optimization

We experimentally performed complete and optimized quantum process tomography of quantum gates implemented on superconducting qubit-based IBM QX2 quantum processor via two constrained convex optimization (CCO) techniques: least squares optimization and compressed sensing optimization. We studied the performance of these methods by comparing the experimental complexity involved and the experimental fidelities obtained. We experimentally characterized several two-qubit quantum gates: identity gate, a controlled-NOT gate, and a SWAP gate. The general quantum circuit is efficient in the sense that the data needed to perform CCO-based process tomography can be directly acquired by measuring only a single qubit. The quantum circuit can be extended to higher dimensions and is also valid for other experimental platforms.

scholarly journals DIAGONAL-UNITARY 2-DESIGN AND THEIR IMPLEMENTATIONS BY QUANTUM CIRCUITS

2013 ◽  
Vol 11 (07) ◽  
pp. 1350062 ◽  
Author(s):  
YOSHIFUMI NAKATA ◽  
MIO MURAO
Keyword(s):  
Algebraic Structure ◽  
Quantum Circuits ◽  
Random Choice ◽  
Unitary Matrices ◽  
Single Qubit ◽  
Classical Procedure ◽  
Random States ◽  
Computational Basis

We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitaryt-designs and present two quantum circuits that implement diagonal-unitary 2-design with the computational basis in N-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary 2-design after applying the gates on all pairs of qubits. The number of required gates is N(N - 1)/2. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact 2-design, but achieves an ϵ-approximate 2-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most O(N2(N + log 1/∊)). We also provide an application of the circuits, a protocol of generating an exact 2-design of random states by combining the circuits with a simple classical procedure requiring O(N) random classical bits.

A Quantum Model of Computation

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Quantum Circuit ◽  
Circuit Model ◽  
Circuit Diagram ◽  
Qubit State ◽  
Quantum Gates ◽  
Quantum Model ◽  
Reversible Circuit ◽  
Logical Qubits ◽  
General Measurement ◽  
Computational Basis

In Section 1.3, we introduced the circuit model of (classical) computation. We restricted attention to reversible circuits since they can simulate any non-reversible circuit with modest overhead. This model can be generalized to a model of quantum circuits. In the quantum circuit model, we have logical qubits carried along ‘wires’, and quantum gates that act on the qubits. A quantum gate acting on n qubits has the input qubits carried to it by n wires, and n other wires carry the output qubits away from the gate. A quantum circuit is often illustrated schematically by a circuit diagram as shown in Figure 4.1. The wires are shown as horizontal lines, and we imagine the qubits propagating along the wires from left to right in time. The gates are shown as rectangular blocks. For convenience, we will restrict attention to unitary quantum gates (which are also reversible). Recall from Section 3.5.3 that non-unitary (non-reversible) quantum operations can be simulated by unitary (reversible) quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A circuit diagram describing a superoperator being implemented using a unitary operator is illustrated in Figure 4.2. In the example of Figure 4.1, the 4-qubit state |ψi⟩= |0⟩⊗ |0⟩⊗ |0⟩⊗ |0⟩ enters the circuit at the left (recall we often write this state as |ψi⟩ = |0⟩|0⟩|0⟩|0⟩ or |ψi⟩ = |0000⟩.) These qubits are processed by the gates U1, U2, U3, and U4. At the output of the circuit we have the collective (possibly entangled) 4-qubit state |ψf⟩. A measurement is then made of the resulting state. The measurement will often be a simple qubit-by-qubit measurement in the computational basis, but in some cases may be a more general measurement of the joint state. A measurement of a single qubit in the computational basis is denoted on a circuit diagram by a small triangle, as shown in Figure 4.1 (there are other symbols used in the literature, but we adopt this one). The triangle symbol will be modified for cases in which there is a need to indicate different types of measurements.R50

scholarly journals Universal quantum computing using single-particle discrete-time quantum walk

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Shivani Singh ◽  
Prateek Chawla ◽  
Anupam Sarkar ◽  
C. M. Chandrashekar
Keyword(s):  
Hilbert Space ◽  
Discrete Time ◽  
Single Particle ◽  
Quantum Walk ◽  
Quantum Gates ◽  
Position Space ◽  
Single Qubit ◽  
Time Quantum ◽  
Universal Quantum ◽  
Qubit States

AbstractQuantum walk has been regarded as a primitive to universal quantum computation. In this paper, we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems by using the operations required to describe the single particle discrete-time quantum walk on a position space. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems.

Computational power of single qubit discrete-time quantum walk

2021 ◽  
Author(s):  
Shivani Singh ◽  
Prateek Chawla ◽  
Anupam Sarkar ◽  
C. M. Chandrashekar
Keyword(s):  
Discrete Time ◽  
Quantum Walk ◽  
Quantum Gates ◽  
Computational Power ◽  
Position Space ◽  
Single Qubit ◽  
Qubit System ◽  
Time Quantum ◽  
Universal Quantum ◽  
Qubit States

Abstract Quantum walk has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of gates on two-and three-qubit system. The idea is to reap the effective Hilbert space of the single qubit and the position space on which it evolves in superposition of position space in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates in the form of engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems.

Both Toffoli and Controlled-NOT need little help to universal quantum computing

2003 ◽  
Vol 3 (1) ◽  
pp. 84-92
Author(s):  
Y-Y Shi
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Circuit ◽  
Single Qubit ◽  
Universal Quantum ◽  
Qubit Gate ◽  
Computational Basis ◽  
Universal Gates

What additional gates are needed for a set of classical universal gates to do universal quantum computation? We prove that any single-qubit real gate suffices, except those that preserve the computational basis. The Gottesman-Knill Theorem implies that any quantum circuit involving only the Controlled-NOT and Hadamard gates can be efficiently simulated by a classical circuit. In contrast, we prove that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its like) are universal for quantum computing. Previously only a generic gate, namely a rotation by an angle incommensurate with \pi, is known to be sufficient in both problems, if only one single-qubit gate is added.

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

An upper bound on the threshold quantum decoherence rate

2004 ◽  
Vol 4 (3) ◽  
pp. 222-228
Author(s):  
A.A. Razborov
Keyword(s):  
Upper Bound ◽  
Random Noise ◽  
Quantum Circuit ◽  
Quantum Decoherence ◽  
Single Qubit ◽  
Constant Rate ◽  
Decoherence Rate

Let $\eta_0$ be the supremum of those $\eta$ for which every poly-size quantum circuit can be simulated by another poly-size quantum circuit with gates of fan-in $\leq 2$ that tolerates random noise independently occurring on all wires at the constant rate $\eta$. Recent fundamental results showing the principal fact $\eta_0>0$ give estimates like $\eta_0\geq 10^{-6}\mbox{--}10^{-4}$, whereas the only upper bound known before is $\eta_0\leq 0.74$.}{In this note we improve the latter bound to $\eta_0\leq 1/2$, under the assumption ${\bf QP}\not\subseteq {\bf QNC^1}$. More generally, we show that if the decoherence rate $\eta$ is greater than 1/2, then we can not even store a single qubit for more than logarithmic time. Our bound also generalizes to the simulating circuits allowing gates of any (constant) fan-in $k$, in which case we have $\eta_0\leq 1-\frac 1k$.

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