Efficiently preparable states of light enable quantum computing by measurement

Quantum information processing permits computational and communications tasks that are not possible through classical means. The most prominent examples are efficient solutions to problems that are believed to be classically intractable, like factoring large numbers (which is relevant for cryptography). A crucial necessity in this task is manipulating quantum mechanical superpositions of quantum states representing the logical 0 and 1, the so-called qubits. There are many ways of generating qubits by appropriate physical systems,¹ each of which has its own advantages and drawbacks.

Light is a promising carrier of quantum information due to the sophistication of optical methods and especially low decoherence, i.e., the destruction of quantum mechanical superpositions by interaction with the environment. Unfortunately, single photons are difficult to create and detect, and interact only weakly with one another. In contrast, light modes with many photons (bright modes) are proving to be of great theoretical and experimental interest. Their nonlinear interactions, in particular, offer new opportunities.² Yet, all the schemes suggested to date require using nonlinear optical elements during computation (online), which is very challenging in experimental setups.

To avoid online interaction, we employ the paradigm of measurement-based quantum computing, which makes it possible to split the problem in two parts: creating a suitable entangled multimode resource state by letting the modes interact, and performing the computation by carrying out local measurements on the individual modes.^3,4 The first step is difficult. But, it does not depend on the quantum algorithm we want to implement, and, therefore, it can be done offline (which simplifies matters greatly).

Figure 1. A two-level atom interacts with an array of optical cavities where coherent states |α〉are fed in. The interaction is governed by the Hamiltonian Ĥ and takes time t, which leads to a unitary evolution . Additional laser pulses transform the atomic state between the interactions by unitary Ω. This prepares the resource states on which computation is performed by homodyne measurements (HD).

A light mode with an arbitrary number of photons ‘lives’ in an infinite-dimensional Hilbert space and is often described by the continuous variables (CVs): position and momentum. Because we ultimately want to process quantum information in the form of qubits, we must encode their 2D Hilbert spaces in the infinite-dimensional spaces of the light modes. This is not merely a technicality but is crucial for effective error detection, correction, and eventually fault tolerance.⁵ The scheme we propose incorporates this requirement in a natural way because we create the resource by letting the modes sequentially interact with a two-level atom representing a qubit. We discovered that such a resource state makes it possible to initialize, process, and read out quantum information using only ‘homodyne’ measurements.⁶ Here, the mode of interest is combined with a strong additional coherent state (the local oscillator) in a Mach-Zehnder interferometer. The photon current is detected on both output ports, and the difference gives the desired result, i.e., the one corresponding to the quantum mechanical position or momentum operator (also called x and p quadratures). These measurements can be carried out in current optical experiments with almost 100% efficiency (unlike essentially any other type of measurement).

We sketch the basic idea for a 1D system where every mode is coupled only to its nearest neighbors. While this setting processes just a single logical qubit, it illustrates the essential points. A resource state can be created by letting an atom sequentially interact with light modes in optical cavities (see Figure 1). The Hamiltonian that describes this interaction is provided by the Jaynes-Cummings model. If the laser is sufficiently detuned with respect to the cavity resonance frequency, the ‘dispersive limit’ can be used, for which the Hamiltonian reads , where is the creation (annihilation) operator of the light mode, is the Pauli z operator in the relevant subspace of the atom, and χ is a parameter characterizing the interaction strength. As an input, we take a coherent state |α〉 (produced by an ordinary laser) in every mode. Together with an additional transformation applied to the atom after each interaction by a laser pulse, this creates a state very similar to the cluster state in a suitable encoding (see Figure 2). This is merely one example, as we have found various other Hamiltonians that also yield resource states.

Figure 2. The result of interaction of an atom with a coherent state in a schematic representation of the probability distribution obtained when measuring the x and pquadratures by homodyne detection. Measurement of x quadrature yields no information about state of the qubit, as the two states are indistinguishable along that axis, and induces a unitary in the correlation space. Measuring the p quadrature enables readout of the qubit information because it discriminates the two states with very high probability. |0〉and |1〉 are the ground and excited states of the atom. |α〉and describe the state of the light before and after interaction with the atom.

The theoretical description of the resource states is provided by a straightforward generalization of the theory of matrix-product states (MPSs) to CV systems.⁷ The matrices in the MPS description are computed from the Hamiltonian of the interaction between the mode and the atom and the initial state of the cavity mode. In this MPS picture,⁸ quantum computation takes place in a 2D correlation space, and every measurement of a single mode leads to the application of a unitary operation, which in the correlation system depends on the measurement outcome. Because of the probabilistic nature of the process, it is necessary to measure various modes until the desired unitary is achieved through a random walk. But we can prove that this is efficiently possible.⁹ Initializing and readout can also be done by measurements (see Figure 2).

We have described a scheme that enables efficient quantum computing using only ordinary laser light, feasible atom-light interaction, and homodyne detection. Consequently, it potentially poses fewer experimental challenges than schemes requiring single-photon sources and detectors. Note that similar interactions can be also realized in a purely optical setting using the cross-Kerr effect, which is the modulation of the relative phase of two laser beams depending on their intensity when passing through a suitable nonlinear crystal. However, achieving substantial nonlinearities is still experimentally daunting.² Future research must address the question of how performance of this system will compare with other approaches when decoherence is included. We are especially interested in investigating photon loss, i.e., what happens when a photon is absorbed or scattered away during the two stages of preparation and computation, as this is the most common source of error when the number of photons is large.