Quantum computation and trapped ions

Ion traps and quantum information

Quantum computation and trapped ions

A fundamental concept in Computer Science is the idea of an inherent mathematical complexity of a problem, quantified in terms of how many elementary steps are necessary to obtain a solution. This concept has been meaningful because a problem's complexity can be shown to be broadly invariant with respect to details of the apparatus or steps employed to solve the problem. However, quantum physics strikes at the heart of this thesis, by providing a new way to solve certain problems. Apparently, fewer elementary steps may be required on a quantum machine, compared with machines obeying classical physics alone. This disturbing clash breaks an elegant separation between Physics and computational complexity, and suggests that something is amiss with either the foundations of Computer Science, or the reality of computation using quantum physics. In this section, we provide perspective on the challenge quantum computation provides to Computer Science, describe several physical models of quantum computation, and present two trapped ion realizations of basic quantum gates. These results, together with the observation that quantum computation can be robust even in the presence of finite noise, support the idea that quantum computation, even with a large number of gates and quantum bits, may be physically realistic.

Quantum computation in perspective

How many elementary mathematical steps are needed to solve a problem? Consider the following examples:

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[2.5ex] \parbox[t]{2.5in}{\raggedright [5.5ex] \parbox[t]{2.5in}{\raggedright Multiply two Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} matrices } [3.5ex] \parbox[t]{2.75in}{\raggedright Given a list of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} numbers, does any subset sum to zero? Example: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,5,-8,6,-3,-9\}} has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+5+6-9-3=0} } [5.5ex] Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -bit number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , is $R$ prime? [3.5ex] \parbox[t]{2.5in}{\raggedright Does an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -bit number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} have a prime factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle < M} ?}

Problem	Number of steps
Add two Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -bit numbers	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim n}
Multiply two Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -bit numbers}	\parbox[t]{2.5in}{\raggedright Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim n^2} using primary school method; $\sim n(\log n)(\log \log n)$ using Sh\"onhage-Strassen method}
	\parbox[t]{2.5in}{\raggedright Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim n^3} using standard method; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim n^{2.376}} using Coppersmith-Winograd method}
	\parbox[t]{2.5in}{\raggedright Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim n 2^{n/2}} using 1974 algorithm; solution is easy to verify}
	\parbox[t]{2.5in}{\raggedright Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim \sqrt{n}} using naive method; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim(\log n)^{12}} using AKS method of 2002}
	\parbox[t]{2.5in}{\raggedright Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim 2^{n^{1/3}}} best known today; solution is easy to verify}

These examples illustrate a natural separation between two kinds of problems: those whose solution can be obtained in a number of steps which grows polynomially with the size of the problem, and those which seem to require an exponential number of steps to reach a decision, but are easily verified. The former are classified as P, and the latter are NP ("nondeterministic polynomial") problems. Illustrating the space of these mathematical problems as follows, one can show that P problems are contained within the class of {\bf NP} problems; moreover, there is a subset of NP-complete problems known to be the hardest of these, such that fast solution of any NP-complete problem would allow fast solution of any other {\bf NP} problem.

Today, the boundary between P and NP problems is increasingly becoming well defined, but it remains to be proven that PFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neq} NP; this is one of the major outstanding problems in Computer Science. Conceptually, these classifications rely on a premise known as Church's Thesis, which, in modern terms, may be stated as \begin{quote} Church's Thesis: Any algorithmic process can be simulated with polynomial resource overhead, using a Turing machine. \end{quote} A Turing machine is a standardized universal computer, conceptually realized with an infinitely long, bi-directional tape, off of which data is accessed, and written by a local head. The head also controls motion of the tape; it contains a finite state machine, with a fixed amount of memory:

Essential to Church's Thesis is the contention that the complexity of a given mathematical problem is a fact which is independent of the Physics of the machine used. This is where quantum computation enters, because apparently, problems such as factoring (eg deciding whether Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} has a factor less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} ), are not in P (as far as is known today), if the Turing machine is constrained to use only classical laws of motion. However, if quantum physics is permitted, the problem can be solved in polynomial time. The class of problems which can be solved, with bounded error, in polynomial time, with a quantum Turing machine, is known as BQP. This class includes P, and today is believed to extend further and include problems outside of P:

Naturally, one of the central issues arising from this realization is the question: Is quantum computation real? Imagine, for example, that there were some law of Physics which placed an absolute maximum on the number of qubits which could be simultaneously entangled. This limit would need to be one of principle, rather than of practice. If such an in-princple limit existed, then quantum computers of arbitrary size would not be possible, thus allowing a sufficiently large classical computer to simulate the quantum machine with polynomial overhead. Then, with this conjectured law of Physics, BQP could become equal to P, rendering meaningless this distinction of quantum versus classical computation (and meanwhile, reducing factoring to a problem in P). On the other hand, if realistic prescriptions for arbitrarily large quantum computers can be given, and no new principles of Physics are found which are inconsistent with their realization, then the foundations of Computer Science would have to be revised. Thus, in the spirit of determining the reality of quantum computation, it is interesting to consider how they might be realized, for example, using trapped ions.

Models of quantum computation

Universal quantum computers can be realized using several models, which differ importantly in the physical resources called for. The most widely prescribed resource set for realizing a quantum computer, known as DiVincenzo's Criteria, require: \begin{itemize}

• Individually addressable qubits, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_1, T_2 \gg T_{gate}}
• Arbitrary unitary transforms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}
• Initial state preparation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\cdots 00{\rangle}}
• Projective measurement in the computational basis

\end{itemize} These requirements are based on the idea of building a {\em quantum circuit} from basic elements, quantum gates, following the theorem that \begin{quote} Theorem: Any $n$-qubit unitary transform Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} can be constructed from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(n^2 4^n)} single qubit gates and two-qubit controlled-{\sc not} gates. \end{quote} Single qubit gates are typically realized, with trapped ion qubits, using laser pulses which perform rotations on the Bloch spheres representing two-level pairs of internal states. The {\sc cnot} gate captures the need for interactions between qubits; it is representative, and not specifically required. The unitary transform implemented by a {\sc cnot} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x{\rangle}|y \rangle \rightarrow |x{\rangle}|x\oplus y{\rangle}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oplus} represents binary addition modulo two. It is equivalent to a controlled-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} gate, also known as a {\sc cphase} gate, up to two Hadamard gates:

\noindent The quantum circuit model can thus be said to require Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}} state qubits, a family of small quantum gates from which arbitrary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} can be constructed, and projective measurements in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1{\rangle}} basis. The teleportation model of quantum computation uses the fact that certain gates can be embedded inside an entangled state, using this circuit:

\noindent Here, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} represents a projective measurement in the Bell basis (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |00{\rangle}\pm |11{\rangle}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |01{\rangle}\pm|10{\rangle}} ), and the "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <} " on the bottom left denotes an entangled state on half of which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is performed. With the appropriate recovery operation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , selected from just two gates, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} , the output state can be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U|\psi{\rangle}} , for input Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi{\rangle}} , and for certain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} . Thus, this model of quantum computation asks for Bell basis measurements, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} gates, and a supply of appropriately entangled states. Another model is given by measurement-based quantum computation, which starts with a specific entangled "cluster" state, for example a square grid of qubits

\noindent in which nearest neighbors are coupled by application of a {\sc cphase} gate. Alternatively, this initial cluster state could possibly be created by evolving a system of coupled spins into the ground state of some Hamiltonian. Quantum computations are then performed by measuring individual qubits in varying bases. These measurements proceed sequentially, for example, from left to right, with earlier measurement results determining the basis used to subsequent measurements. Thus, this model of quantum computation requires an initial "cluster" entangled states, and single qubit measurements in arbitrary bases. The quantum circuit model is appealing to a wide range of physical implementations, because the most difficult element to realize is a two-qubit gate such as the {\sc cnot} or {\sc cphase}. The teleportation model is used particularly in quantum computation with linear optics, partly due to the ease with which entangled photon pairs can be created, and mainly due to the ability to separate gates from qubits containing data until the right time. The measurement based model has been applied to optical qubits, but is mostly likely of future interest to schemes such as optical lattices of atoms, which can naturally realize the requisite entangled cluster state. Clearly, it helps to choose the model which best matches naturally available resources for realistic experimental realization of quantum computation.

The Cirac-Zoller gate

Let us now consider how the {\sc cnot} gate could be realized with qubits represented by trapped ions. Suppose we have an idealized system of two trapped ions, confined together in single one-dimensional harmonic potential, sharing a common center-of-mass vibrational mode of frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} , much like a single molecule:

Let the qubits be represented by spin states in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} level of the ion, and suppose the ion has an additional "auxiliary" level (which could be another Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_F} level, or something in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} state manifold), that is selectively accessible by an optical transition from $|1{\rangle }$:

The red sideband of this transition from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1 \rangle = |e{\rangle}} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |{\rm aux}{\rangle}} couples $|e,1{\rangle }$ to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |{\rm aux},0{\rangle}} , but not Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e,0{\rangle}} :

\noindent Recall that for a rotation operator

$R_{x}(\theta )=\cos {\frac {\theta }{2}}-i\sigma _{x}\sin {\frac {\theta }{2}}\,,$

a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} pulse flips the qubit, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_x(\pi) = -i\sigma_x} , but a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} pulse leaves it with an overall $-1$ sign, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_x(2\pi) = -1} . Thus, a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} pulse on the red sideband of the transition from $|e{\rangle }$ to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |{\rm aux}{\rangle}} , assuming the motion starts in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}} state, results in the transform

${\begin{array}{rcl}|g,0\rangle &\rightarrow &|g,0{\rangle }\\|g,1\rangle &\rightarrow &|g,1{\rangle }\\|e,0\rangle &\rightarrow &|e,0{\rangle }\\|e,1\rangle &\rightarrow &-|e,1{\rangle }\,.\end{array}}$

This is the unitary transform of the {\sc cphase} gate. A controlled-phase gate can thus be realized between the two ions by initializing the shared center-of-mass phonon state to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}} , swapping a qubit from ion 2 into the phonon, performing the {\sc cphase} gate just described with pulses on ion 1, then swapping the phonon back into ion 2. Swapping the phonon and internal state can be done using a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} pulse on the red sideband of the carrier transition:

Thus, the overall procedure for this Cirac-Zoller gate is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} {\rm initialize} &\rightarrow& (a|g{\rangle}+b|e \rangle )(c|g{\rangle}+d|e \rangle )|0{\rangle} \\ {R_{eg2}^{RSB}(\pi)} &\rightarrow& (a|g{\rangle}+b|e \rangle )|g \rangle (c|0{\rangle}+d|1 \rangle ) \\ {R_{aux1}^{RSB}(2\pi)} &\rightarrow& ac|gg0 \rangle + ad|gg1 \rangle + bc|eg0{\rangle} - bd|eg1{\rangle} \\ {R_{eg2}^{RSB}(\pi)} &\rightarrow& \left[ { ac|gg \rangle + ad|ge \rangle + bc|eg{\rangle} - bd|ee \rangle } \right] |0{\rangle} \,. \end{array}}

This implements a {\sc cphase} gate, from which a {\sc cnot} can be obtained by prepending and post-pending Hadamard gates on the target qubit. The major limitations of this scheme are that (1) the phonon state must be initialized to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}} , so that good resolved sideband cooling is necessary, and (2) the speed of the gate is limited to $\nu ^{-1}$, due to the pulse widths required for the sideband excitations.

The geometric phase gate

Some of the limitations of the Cirac-Zoller gate can be circumvented using a different method for realizing a {\sc cphase} gate, based on geometric phases. Recall that using a "moving standing wave," a state dependent force which maps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g,0{\rangle}\rightarrow |g,0{\rangle}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e,0 \rangle \rightarrow |g,\alpha{\rangle}} can be realized:

\noindent This implements a displacement operator $D(\alpha )$ on the phonon mode. Two back-to-back displacements give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D(\alpha) D(\beta) = D(\alpha+\beta) e^{i{\rm Im}(\alpha\beta^*)} \,, }

where the extra phase factor comes from the path enclosing some area in the phase space of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm Re}(\alpha)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm Im}(\alpha)} plane. Conceptually, this extra phase could be used to provide a qubit state selective phase, if the phonon state is displaced in a path which starts from and returns to the same point in phase space. For example, displacements could be arranged to move the phonon around a circle, with the area selected to map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g{\rangle}\rightarrow |g{\rangle}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e{\rangle}\rightarrow -|e{\rangle}} :

Furthermore, this idea can be generalized to impart a state-selective phase to two ions, in the following way. Consider two ions in a moving standing wave of detuning ${\sqrt {3}}\nu$, such that the frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} center-of-mass mode is left unexcited, and but the frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{3}\nu} stretch mode is excited:

Specifically, suppose that the force is imparted by a laser beam selected to act on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e{\rangle}} but not on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g{\rangle}} , so that if the internal state is $|e{\rangle }$ the force is large, and if the internal state is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |g{\rangle}} the force is small. These forces thus do not excite the stretch mode if both ions are in the same state, either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |ee{\rangle}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |gg{\rangle}} . However, if the ions have opposite spin, either $|eg{\rangle }$ or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |ge{\rangle}} , then the differential force between the two ions excites their stretch mode:

Applying a force tuned to displace the two ions around a circle in phase space can thus give a selective phase factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} to $|eg{\rangle }$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |ge{\rangle}} , while leaving Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |gg{\rangle}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |ee{\rangle}} with no phase change:

This implements the unitary transform

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & 0 & 0 & 1 \\ \end{matrix} \right ] \approx e^{i(\pi/4) \sigma_{z1} \sigma_{z2}} \,, }

where the equality is up to an overall (and irrelevant) global phase factor. Adding two single qubit rotations gives a controlled-{\sc phase} gate:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{z1}(\pi/2) R_{z2}(\pi/2) U = \mbox{\sc cphase} \,, }

and adding two additional Hadamard gates gives a {\sc cnot}.

Quantum computation with noisy gates

How robust is quantum computation in the presence of noise? It turns out a similar question was asked about classical computers, early in their history. This motivated John von Neumann to study the possibility of the synthesis of reliable organisms from unreliable parts, and inspired by biological systems, he proved a theorem which has evolved into the following: \begin{quote} Theorem: A circuit containing $N$ error-free gates can be simulated with probability of error less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} , using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim N \log(N/\epsilon)} faulty gates, which fail with probability at most Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , as long as $p$ is less than some threshold Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{th}} . \end{quote} Note that the failure probability threshold Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{th}} is {\em independent} of the circuit size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} , and the desired reliability $\epsilon$. This is because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{th}} is actually determined, in principle, by properties of error correction codes, not by the ideal system being simulated. For classical circuits and using classical error correction codes, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{th}} can be as high as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/6} . Today, an analogous theorem which applies to quantum circuits is known. However, because of the additional kinds of errors which can happen to qubits, versus bits, quantum error correction codes are correspondingly more complex, and thus $p_{th}$ for quantum computation is believed to be in the range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-4}} to perhaps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-3}} . The fact that such thresholds even exist is remarkable, for quantum computation. Their existence implies that below a certain, constant level of noise, arbitrarily large quantum computations can be performed with a resource overhead which grows very slowly with respect to the size of the computational task. Even more remarkably, gate error probabilities of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-4}} may not be unreasonable for realistic implementations of quantum computers, such as trapped ions. This can be seen from careful analysis of the Cirac-Zoller and geometric phase gates, for example, in which the coherence of the intermediate phonon state is the most limiting element. Careful experiments should thus be done to quantify achievable coherence times, and to predict eventual quantum gate error probabilities with reasonable extrapolations about technological capabilities for systems with large numbers of qubits.