Conditional Quantum Dynamics and Logic Gates

Richard Jozsa

May 3, 1999

Abstract

Quantum logic gates provide fundamental examples of conditional quantum dynamics. They could form the building blocks of general quantum information processing systems which have recently been shown to have many interesting non–classical properties. We describe a simple quantum logic gate, the quantum controlled–NOT, and analyse some of its applications. We discuss two possible physical realisations of the gate; one based on Ramsey atomic interferometry and the other on the selective driving of optical resonances of two subsystems undergoing a dipole– dipole interaction.

PACS: 03.65.Bz, 03.75.Dg, 89.70.+c

The fact that quantum–mechanical processes in principle allow new types of information processing has been known for almost a decade [1, 2]. Bennett and Wiesner have shown that the capacity of quantum channels can be doubled [3] and recent progress in quantum complexity theory [4] indicates that the computational power of quantum computers exceeds that of Turing machines. Hence the experimental realisation of such processes is a most interesting issue. In this paper we concentrate on the basic constituents of any quantum information processing device, namely quantum logic gates. We wish to stress the apperance of a conditional quantum dynamics, in which one subsystem undergoes a coherent evolution that depends on the quantum state of another subsystem. The unitary evolution operator for the combined system has the form

U = |0⟩⟨0| ⊗ U_{0} + |0⟩⟨0| ⊗ U_{1} + … + |k⟩⟨k| ⊗ U_{k}, (1)

where the projectors refer to quantum states of the control subsystem and the unitary operations U_{i} are performed on the target subsystem. The simplest non–trivial operation of this sort is the quantum controlled–NOT. We describe this gate, analyse some of its applications and discuss physical realisations. The classical controlled–NOT gate is a reversible logic gate operating on two bits ϵ_{1} and ϵ_{2}; ϵ_{1} is called the control bit and ϵ_{2} the target bit. The value of ϵ_{2} is negated if ϵ_{1} = 1, otherwise ϵ_{2} is left unchanged. In both cases the control bit ϵ_{1} remains unchanged. We define the quantum controlled–NOT gate C_{12} as that which effects the unitary operation on two qubits (two–state quantum systems), which in a chosen orthonormal basis {|0⟩, |1⟩} in H_{2} reproduces the controlled–NOT operation:

|ϵ_{1}⟩|ϵ_{2}⟩ ⟶^{C_{12}} |ϵ_{1}⟩|ϵ_{1} ⊕ ϵ_{2}⟩,  (2)

where ⊕ denotes addition modulo 2. Here and in the following the first subscript of C_{ij} always refers to the control bit and the second to the target bit. Thus for example C_{21} performs the unitary operation defined by:

|ϵ_{1}⟩|ϵ_{2}⟩ ⟶^{C_{21}} |ϵ_{1} ⊕ ϵ_{2}⟩|ϵ_{2}⟩.  (3)

The quantum controlled–NOT must be distinguished from the classical controlled–NOT which is performable on existing computers. The quantum controlled–NOT is a coherent operation on quantum states of the two qubits. The unitary operation defined by (2) is not the only one which reproduces the classical controlled–NOT on the computation basis states |0⟩ and |1⟩. We may introduce extra phases and the most general such quantum operation is:

|ϵ_{1}⟩|ϵ_{2}⟩ ⟶ exp(iθ_{ϵ_{1}ϵ_{2}})|ϵ_{1}⟩|ϵ_{1} ⊕ ϵ_{2}⟩.  (4)

This phase would be irrelevant to classical operations but gives rise to a family of inequivalent quantum gates.

Equations (2) or (3) define the gate C_{12} with respect to a specific basis, the computation basis {|0⟩, |1⟩}. It is also useful to consider generalisations of C_{12} which have the analogous effect on the control and target bits in bases that are different from the computation basis and possibly from each other. For example C_{12} with respect to the basis \{\frac{1}{\sqrt{2}}(|0⟩ ± |1⟩)\} (for both qubits) is easily shown to be identical to C_{21} with respect to the basis {|0⟩, |1⟩} i.e. the roles of the qubits are reversed by this simple change of basis. In the following we always use the computation basis unless otherwise stated.

The quantum controlled–NOT gate has a variety of interesting properties and applications:

(1) C_{12} transforms superpositions into entanglements

C_{12} : (a|0⟩ + b|1⟩)|0⟩ ⟷ a|0⟩|0⟩ + b|1⟩|1⟩. (5)

Thus it acts as a measurement gate because if the target bit ϵ_{2} is initially in state |0⟩ then this bit together with the gate amount to an apparatus that performs a perfectly accurate non–perturbing (quantum non– demolition [5]) measurement of ϵ_{1}.

(2) This transformation of superpositions into entanglements can be reversed by applying the same controlled–NOT operation again. Hence it can be used to implement the so–called Bell measurement [6] on the two bits by disentangling the Bell states. From the four Bell states we get four product states:

C_{12} \frac{1}{\sqrt{2}}(|0⟩|0⟩ ± |1⟩|1⟩) = \frac{1}{\sqrt{2}}(|0⟩ ± |1⟩)|0⟩, (6)
C_{12} \frac{1}{\sqrt{2}}(|0⟩|1⟩ ± |1⟩|0⟩) = \frac{1}{\sqrt{2}}(|0⟩ ± |1⟩)|1⟩. (7)

Thus the Bell measurement on the two qubits is effected by a sequence of two independent two– dimensional measurements: in the computation basis for the target qubit and in the basis \{\frac{1}{\sqrt{2}}(|0⟩ ± |1⟩)\} for the control qubit. The realisation of the Bell measurement is the main obstacle to the practical implementation of quantum teleportation [7] and dense quantum coding [3].

(3) Quantum state swapping can be achieved by cascading three quantum controlled–NOT gates:

C_{12}C_{21}C_{12}|ψ⟩|ϕ⟩ = |ϕ⟩|ψ⟩,  (8)

for arbitrary states |ψ⟩ and |ϕ⟩ (see also [8]).

(4) The quantum controlled–NOT gate may also be used to swap distantly separated states in the presence of a channel carrying only classical information. This is in contrast to the state swapping described above which requires the gate to be applied to the two states as inputs, so that they cannot be distantly separated at the time. Suppose that Alice and Bob, distantly separated, have states |α⟩ in H_{0} and |β⟩ in H_{5} respectively which they wish to swap (the identities of the states are assumed to be unknown to Alice and Bob). To achieve this they will need, on a previous occasion when they were close together or had access to a quantum communication channel, to have shared two pairs of qubits, one in the state \frac{1}{\sqrt{2}}(|0⟩|0⟩ + |1⟩|1⟩) in H_{1} ⊗ H_{3} and the other in the same maximally entangled state in H_{2} ⊗ H_{4}. States in H_{0},H_{1}, H_{2} are localised near Alice and states in H_{3}, H_{4}, H_{5} are localised near Bob. Let M denote a complete measurement in the computational basis {|0⟩, |1⟩}.

To swap |α⟩ and |β⟩ Alice and Bob carry out the following protocol. Step 1: Alice performs C_{10} and then C_{02} while Bob performs C_{54} and then C_{35}. Step 2: Alice measures M in H_{2} and Bob measures M in H_{4}. Each communicates the result (one bit of information) to the other participant. If the results are the same, go to step 3. If the results are different, Alice and Bob negate all bits in their possession i.e. apply the unitary operation

( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

to each particle. Step 3: Alice applies the rotation

\frac{1}{\sqrt{2}} ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} )

in H_{1} and Bob does the same in H_{3}. Step 4: Alice performs the measurement M in H_{1} and Bob performs it in H_{3}. Each communicates the result to the other. If the results agree then the states will have been swapped. If the results differ then Alice applies the unitary transformation

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

to H_{0} and Bob does the same to H_{5} after which the states will have been swapped. A related process has been described by Vaidman[9].

It is interesting to compare the protocol to quantum teleportation [7], in which Alice and Bob initially share one maximally entangled pair and Alice is able to transfer an arbitrary state |ξ⟩ to Bob by sending him only two classical bits of information. Thus using the same resources as in our protocol viz. sharing two entangled pairs and each participant sending two bits to the other, we may alternatively swap the states |α⟩ and |β⟩ by performing two teleportations (for the two directions of transfer.) However, the process above cannot be separated into two successive transfers. The remarkable feature of all these processes is that in the presence of shared entanglement an arbitrary state |ξ⟩ may be transferred as a result of sending only a few bits of classical information despite the fact that |ξ⟩ depends on two continuous parameters corresponding to an infinite amount of classical information.

The quantum controlled–NOT gate is not a universal gate. However, together with relatively trivial single–qubit operations it forms an adequate set of quantum gates i.e. the set from which any quantum gate may be built [10]. Thus the conditional dynamics of the quantum controlled–NOT type would in realistic technologies be sufficient to construct any quantum information processing device. Universal two–qubit quantum gates based on similarly controlled dynamics are described in [11].

In the following we outline two possible experimental realisations of the quantum controlled–NOT gate. We do not wish to suggest that these particular technologies are destined to yield practicable devices. They do, however, serve to illustrate physical considerations that would bear on the building of such devices in any technology.

The first technology is that of Ramsey atomic interferometry [12, 13, 14, 15], the second is based on the selective driving of optical resonances of two qubits undergoing a dipole–dipole interaction [16]. In the Ramsey atomic interferometry method the target qubit is an atom with two circular Rydberg states |ϵ_{2}⟩, where ϵ_{2} = 0, 1; the control qubit is the quantized electromagnetic field in a high–Q cavity C. The field in the cavity contains at most one photon of a particular mode so it can be viewed as a two state system with the vacuum state |0⟩, and the one– photon state |1⟩ as the basis. The cavity C is sandwiched between two auxiliary microwave cavities R_{1} and R_{2} in which classical microwave fields produce π/2 rotations of the Bloch vector of an atom passing through at a given speed,

|ϵ_{1}⟩_{field}|ϵ_{2}⟩_{atom} ⟶ |ϵ_{1}⟩_{field} \frac{1}{\sqrt{2}}(|ϵ_{2}⟩+(−1)^{ϵ_{2}}e^{iα}|1−ϵ_{2}⟩)_{atom}.

(9)

where the phase factor α is different for the two cavities R_{1} and R_{2}. In the central cavity C, a dispersive interaction with the quantized field introduces phase shifts which depend on the state of the atom |ϵ_{2}⟩ and on the number of photons in the cavity |ϵ_{1}⟩. The interaction conserves the number of photons in the cavity.

|ϵ_{1}⟩_{field}|ϵ_{2}⟩_{atom} ⟶ exp(i(−1)^{1−ϵ_{2}}(ϵ_{1}+ϵ_{2})θ)|ϵ_{1}⟩_{field}|ϵ_{2}⟩_{atom},

(10)

where θ, the phase shift per photon, can be tuned to be π ( θ depends on the time taken for the atom to cross C and the atom–field detuning).

The overall process can be viewed as a sequence: half–flipping in R_{1}, phase shifts in C, and half– flipping in R_{2}. Depending on the phase shifts the second half–flipping can either put the atom back into its initial state or flip it completely into the orthogonal state. The whole interferometer can be adjusted so that when the atom passes successively through the cavities R_{1}, C, and R_{2} the two qubits, i.e. the field and the atom, undergo the transformation

|ϵ_{1}⟩_{field}|ϵ_{2}⟩_{atom} ⟶ |ϵ_{1}⟩_{field}|ϵ_{1} ⊕ ϵ_{2}⟩_{atom}. (11)

The state of the field in C can also be transfered from (and to) an auxiliary Rydberg atom tuned to the resonant frequency of the cavity so that it undergoes a resonant rather than a dispersive interaction in C. This process allows the creation of a gate acting on two qubits of the same type, i.e. two Rydberg atoms rather than a field and an atom. Davidowich et al. [13] have shown how to use the Ramsey interferometry for quantum teleportation. Their experimental setup effectively contains conditional dynamics of the type we have been discussing, which has much wider application in quantum information processing than merely quantum teleportation. The practical realisation of the quantum controlled–NOT gate can be achieved with a modification of the experiments as described in [13, 14, 15]. The typical resonant frequency would be of the order ∼ 2 × 10^{10} Hz, the atom–field interaction time in the cavity ∼ 3×10^{−5} s, and the cavity field lifetime can be made as long as ∼ 0.5 s.

The most difficult part of the experimental realisation is likely to be the preparation of a single atom. This is usually done by preparing an atomic beam with a very low probability of finding a single atom in the beam; consequently finding two atoms in the beam is even less probable. From our point of view the drawback of this method is that it forces a trade– off between the probability that precisely one atom (as required) interacted with the field on a given run, and the reliability of the gate. Although our example has focused on microwave cavities, experimental realisations in the optical regime could also be considered [15].

Our second proposal for the implementation of the quantum controlled–NOT gate relies on the dipole– dipole interaction between two qubits. For the purpose of this model the qubits could be either magnetic dipoles, e.g. nuclear spins in external magnetic fields, or electric dipoles, e.g. single–electron quantum dots in static electric fields. Here we describe the model based on interacting quantum dots, but mathematically the two cases are isomorphic.

Two single–electron quantum dots separated by a distance R are embedded in a semiconductor. Let us consider the ground state and the first excited state of each dot as computation basis states |0⟩ and |1⟩. The first quantum dot, with resonant frequency ω_{1}, will act as the control qubit and the second one, with resonant frequency ω_{2}, as the target qubit. In the presence of an external static electric field, which can be turned on and off adiabatically in order to avoid transitions between the levels, the charge distribution in the ground state of each dot is shifted in the direction of the field whilst in the first excited state the charge distribution is shifted in the opposite direction (the quantum–confined Stark effect)[17], see Fig. 1. In the simple model in which the state of the qubit is encoded by a single electron per quantum dot, we can choose coordinates in which the dipole moments in states |0⟩ and |1⟩ are ±d_{i}, where i = 1, 2 refers to the control and target dot respectively. For the sake of clarity, we are presenting the idea using a slightly simplified model. A more elaborate model would take into account holes in the valence band of the semiconductors. The state of a qubit would be determined by excitons of different energies.

The electric field from the electron in the first quantum dot may shift the energy levels in the second one (and vice versa), but to a good approximation it does not cause transitions. That is because the total Hamiltonian

Ĥ = Ĥ_{1} + Ĥ_{2} + V̂_{12}  (12)

is dominated by a dipole–dipole interaction term V̂_{12} that is diagonal in the four–dimensional state space spanned by eigenstates {|ϵ_{1}⟩, |ϵ_{2}⟩} of the free Hamiltonian Ĥ_{1} + Ĥ_{2}, where ϵ_{1} and ϵ_{2} range over 0 and 1. Specifically,

(Ĥ_{1} + Ĥ_{2})|ϵ_{1}⟩|ϵ_{2}⟩ = h̄(ϵ_{1}ω_{1} + ϵ_{2}ω_{2})|ϵ_{1}⟩|ϵ_{2}⟩, (13)

and

V̂_{12}|ϵ_{1}⟩|ϵ_{2}⟩ = (−1)^{ϵ_{1}+ϵ_{2}}h̄ω̄|ϵ_{1}⟩|ϵ_{2}⟩,  (14)

where

ω̄ = −\frac{d_{1}d_{2}}{4πϵ_{0}R^{3}} .  (15)

As shown in Fig. 2, it follows that due to the dipole–dipole interaction the resonant frequency for transitions between the states |0⟩ and |1⟩ of one dot depends on the neighbouring dot’s state. This is the conditional quantum dynamics we are seeking. The resonant frequency for the first dot becomes ω_{1} ± ω̄ according as the second dot is in state |0⟩ or |1⟩ respectively. Similarly the second dot’s resonant frequency becomes ω_{2} ± ω̄, depending on the state of the first dot. Thus a π–pulse at frequency ω_{2} + ω̄ causes the transition |0⟩ ↔ |1⟩ in the second dot if and only if the first dot is in state |1⟩.

For such processes to be useful for quantum information processing, the decoherence time must be greater than the time scale of the optical interaction (see for example [18]). The decoherence time depends partly on the modification of the confining potential due to phononic excitations. There is also a quantum electrodynamic contribution due to coupling to the vacuum modes. For resonant frequencies in the infrared regime, this can be estimated at ∼ 10^{−6} s. Impurities and thermal vibration (phonons) can reduce the lifetime further to ∼ 10^{−9} s or even worse, but in principle their effects can be minimized by a more precise fabrication technology and by cooling the crystal. The optical interaction time–scale can be approximated by the length of the π–pulse (∼ 10^{−9} s). The length of the pulse is not so much restricted by the current technology as by the requirement for the π–pulse to be monochromatic and selective enough; this restricts the length of the pulse to being greater than the inverse of the pulse carrier frequency and the inverse of the dipole–dipole interaction coupling constant (1/ω̄ ∼ 10^{−12} s in our model). This model may be more difficult to implement than the one based on the Ramsey atomic interferometry, but once it is implemented it is likely allow for quantum gates to be integrated more easily into complex quantum circuits, as required for more general quantum information processing.

Acknowledgements

The authors thank B.G. Englert, S. Haroche, H.J. Kimble, H. Mabuchi, G. Mahler, J–M. Raimond, H. Walther for discussions and comments. This work has been partially supported by the NIST Advanced Technology Program. A. B. gratefully acknowldeges the financial support of the Berrow’s Fund at Lincoln College (Oxford). Work by A. E. is supported by the Royal Society, London.

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Captions

Figure 1: Charge density in the quantum well in the direction x of the applied field. A dipole moment is induced when the electric field is turned on (B), but is zero without the electric field (A).

Figure 2: (a) Energy levels of two quantum dots without and with the coupling induced by the presence of a static electric field E_{0}. (b) Resonance spectrum of the two quantum dots. The dotted line shows the wavelength for which the two dots act as a controlled–NOT gate, with the first dot being the control qubit and the second the target qubit.