It is shown that Deutsch’s algorithm and the Deutsch-Jozsa algorithm for two and three qubits can be efficiently implemented on a classical computer. This is done by analysing the structure of the oracles contained in these algorithms, followed by an implementation in the toy theory proposed by Robert W. Spekkens which uses 2 classical bits to simulate a qubit. Further discussion is given about how to analyse, and possible implementations, of the Deutsch-Jozsa for higher number of qubits in the toy theory.It is shown that Deutsch’s algorithm and the Deutsch-Jozsa algorithm for two and three qubits can be efficiently implemented on a classical computer. This is done by analysing the structure of the oracles contained in these algorithms, followed by an implementation in the toy theory proposed by Robert W. Spekkens which uses 2 classical bits to simulate a qubit. Further discussion is given about how to analyse, and possible implementations, of the Deutsch-Jozsa for higher number of qubits in the toy theory.

Jan 1, 20140 citations

Single-step implementation of high-fidelity n -bit Toffoli gates

S. E. Rasmussen, Koen Groenland, R. Gerritsma +2 more

The family of $n$ -bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary $n$ -bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to $n$ other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely, strongly coupling $n+1$ qubits and a driving frequency that scales with $n$ , may break down for large systems. Still, our protocol could enhance the capabilities of intermediate-scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-not gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.The family of $n$-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary $n$-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to $n$ other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely, strongly coupling $n+1$ qubits and a driving frequency that scales with $n$, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate-scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-not gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.

arXiv:1910.07548Physical Review AFeb 7, 202070 citations

Computation with Qubits

WORLD SCIENTIFIC eBooksJul 15, 20200 citations

Quantum Computing Fundamentals: Beyond Classical Bits

Swati Karni, Arun Kumar Rajamandrapu

Springer eBooksJan 1, 20260 citations

Experimental Approximation of the Jones Polynomial with One Quantum Bit

Gina Passante, Osama Moussa, Colm A. Ryan +1 more

We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.

arXiv:0909.1550Physical Review LettersDec 18, 200940 citations

The family of

n

-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary

n

-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to

n

other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely, strongly coupling

n+1

qubits and a driving frequency that scales with

n

, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate-scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-not gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.