Optimal phase estimation for qubit mixed states

Physical Review A, 2005

We present the optimal phase estimation for qubits in mixed states, for an arbitrary number of qubits prepared in the same state.

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N , we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N . We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.

2005

We study the estimation problem of a general mixed qubit state with protocols based on local measurements. We obtain the asymptotic bound of the fidelity for these settings and show that they do not attain the optimal joint measurement bound. We present an explicit protocol that uses classical communication in a very efficient way and saturates the local bound. We also analyze the case of mixed states known to lie on an equatorial plane of the Bloch sphere.

Physics Letters A, 1998

The problem of estimating a generic phase-shift experienced by a quantum state is addressed for a generally degenerate phase shift operator. The optimal positive operator-valued measure is derived along with the optimal input state. Two relevant examples are analyzed: i) a multi-mode phase shift operator for multipath interferometry; ii) the two mode heterodyne phase detection.

Physical Review Letters, 2007

We address the problem of estimating the phase given N copies of the phase-rotation gate u . We consider, for the first time, the optimization of the general case where the circuit consists of an arbitrary input state, followed by any arrangement of the N phase rotations interspersed with arbitrary quantum operations, and ending with a general measurement. Using the polynomial method, we show that, in all cases where the measure of quality of the estimate for depends only on the difference ÿ , the optimal scheme has a very simple fixed form. This implies that an optimal general phase estimation procedure can be found by just optimizing the amplitudes of the initial state.

Lecture Notes in Computer Science, 2020

In the near-term, the number of qubits in quantum computers will be limited to a few hundreds. Therefore, problems are often too large and complex to be run on quantum devices. By distributing quantum algorithms over different devices, larger problem instances can be run. This distributing however, often requires operations between two qubits of different devices. Using shared entangled states and classical communication, these operations between different devices can still be performed. In the ideal case of perfect fidelity, distributed quantum computing is a solution to achieving scalable quantum computers with a larger number of qubits. In this work we consider the effects on the output fidelity of a quantum algorithm when using noisy shared entangled states. We consider the quantum phase estimation algorithm and present two distribution schemes for the algorithm. We give the resource requirements for both and show that using less noisy shared entangled states results in a higher overall fidelity.

Journal of Physics A: Mathematical and Theoretical, 2007

We address the problem of estimating the phase φ given N copies of the phase rotation u φ within an array of quantum operations in finite dimensions. We first consider the special case where the array consists of an arbitrary input state followed by any arrangement of the N phase rotations, and ending with a POVM. We optimize the POVM for a given input state and fixed arrangement. Then we also optimize the input state for some specific cost functions. In all cases, the optimal POVM is equivalent to a quantum Fourier transform in an appropriate basis. Examples and applications are given.

Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring symmetry in the error definitions, an exact formula can be found. This new approach may also have value in solving other related problems in quantum computing, where an expected error is calculated. Expressions for two special cases of the formula are also developed, in the limit as the number of qubits in the quantum computer approaches infinity and in the limit as the extra added qubits to improve reliability goes to infinity. It is found that this formula is useful in validating computer simulations of the phase estimation procedure and in avoiding the overestimation of the number of qubits required in order to achieve a given reliability. This formula thus brings improved precision in the design of quantum computers.

2008

Recently, the field of unconventional computing has witnessed a huge research effort to solve the problem of the assumed power of computers operating purely according to the laws of quantum physics. Quantum computing can be seen as a special intermediate case between digital and real analog computing. Importantly, there is a threshold theorem for error correction, as opposed to the pure analog case. Alternatively, quantum computing can be viewed as generalized probabilistic computing, where non-negative real probabilities are replaced with complex amplitudes. The main new resources are quantum mechanical phenomena such as state superposition, interference and entanglement. Superposition together with interference provide a special kind of parallelism, while entanglement, especially when spatially shared, supports unique means of communication. One of the most important theoretical result is a proof by Bernstein and Vazirani (1993) that there is an oracle relative to which there is a language that can be efficiently accepted by a quantum Turing machine, but cannot be efficiently accepted by a bounded-error probabilistic Turing machine. The problem which was considered is called Recursive Fourier Sampling and the proposed quantum algorithm gives a quasipolynomial speedup, O(n) versus O(n log n). Next, Abrams and Lloyd showed that a quantum computer can efficiently simulate manybody quantum systems having a local Hamiltonian. An additional (informal) evidence of the assumed power of a quantum computer is a bounded-error quantum polynomial time algorithm for large integer factoring. The Abrams-Lloyd algorithm is potentially the most useful quantum algorithm known so far, and if a quantum computer is ever built, it will revolutionize quantum chemical calculations. Thus there is a growing consensus regarding investments into the experimental quantum computing. In this thesis, attention is paid to small experimental testbed applications with respect to the quantum phase estimation algorithm, the core approach for finding energy eigenvalues. An iterative scheme for quantum phase estimation (IPEA) is derived from the Kitaev phase estimation, a study of robustness of the IPEA utilized as a few-qubit testbed application is performed, and an improved protocol for phase reference alignment is presented. Additionally, a short overview of quantum cryptography is given, with a particular focus on quantum steganography and authentication.

New Journal of Physics

We introduce a new statistical and variational approach to the phase estimation algorithm (PEA). Unlike the traditional and iterative PEAs which return only an eigenphase estimate, the proposed method can determine any unknown eigenstate-eigenphase pair from a given unitary matrix utilizing a simplified version of the hardware intended for the Iterative PEA (IPEA). This is achieved by treating the probabilistic output of an IPEA-like circuit as an eigenstate-eigenphase proximity metric, using this metric to estimate the proximity of the input state and input phase to the nearest eigenstate-eigenphase pair and approaching this pair via a variational process on the input state and phase. This method may search over the entire computational space, or can efficiently search for eigenphases (eigenstates) within some specified range (directions), allowing those with some prior knowledge of their system to search for particular solutions. We show the simulation results of the method with the Qiskit package on the IBM Q platform and on a local computer.