Minimal ancilla mediated quantum computation
© Proctor and Kendon; licensee Springer on behalf of EPJ. 2014
Received: 10 April 2014
Accepted: 13 August 2014
Published: 4 September 2014
Schemes of universal quantum computation in which the interactions between the computational elements, in a computational register, are mediated by some ancillary system are of interest due to their relevance to the physical implementation of a quantum computer. Furthermore, reducing the level of control required over both the ancillary and register systems has the potential to simplify any experimental implementation. In this paper we consider how to minimise the control needed to implement universal quantum computation in an ancilla-mediated fashion. Considering computational schemes which require no measurements and hence evolve by unitary dynamics for the global system, we show that when employing an ancilla qubit there are certain fixed-time ancilla-register interactions which, along with ancilla initialisation in the computational basis, are universal for quantum computation with no additional control of either the ancilla or the register. We develop two distinct models based on locally inequivalent interactions and we then discuss the relationship between these unitary models and the measurement-based ancilla-mediated models known as ancilla-driven quantum computation.
The original theoretical setting for quantum computation is the gate model  in which a global unitary that acts on a register of qubits, which computes the solution to some problem, is decomposed into a sequence of fundamental gates that are applied to the register. As in classical computation, it is desirable for these fundamental gates to be members of some finite and universal gate set, from which any global unitary can be composed up to arbitrary accuracy. There has been extensive research on such universal sets, and a significant example is the set composed of any entangling gate in conjunction with any universal set of single-qubit unitaries [2, 3]. Furthermore, almost any two-qubit entangling gate is universal on its own provided that it can be applied to arbitrary pairs of qubits [4, 5]. These results are of significant theoretical importance for the understanding of quantum computation.
However, the physical implementation of these models requires direct interactions between arbitrary pairs of register qubits and, often, direct application of single-qubit rotations and measurements. This is a huge practical challenge and most experimentally implemented or proposed schemes mediate the required multi-qubit interactions using some ancillary system. An example of such an ancilla-mediated scheme is the original Cirac-Zoller ion-trap gate , where the ancilla in this case is the collective quantized motion of the ions. Further examples include superconducting qubits coupled to nitrogen-vacancy (NV) centres [7–10] or transmission line resonators [11, 12], spin qubits coupled via ancillary photonic qubits [13, 14] and the coupling of a Cooper-pair box to a micro-mechanical resonator . It is therefore of both practical and theoretical interest to study the effect of incorporating the ancillary system into the computational model. Indeed, those gates that have been shown to be universal in a direct implementation of the circuit model cannot in general be utilised to implement quantum computations entirely mediated via an ancilla. We will refer to schemes in which all the multi-qubit interactions are mediated via some ancillary system as ancilla-mediated quantum computation (AMQC).
An extensively researched model of AMQC is quantum bus (qubus) computation [16–21]. This model employs a field-mode ancilla to mediate two-qubit gates on pairs of register qubits with the interaction between the ancilla and a register qubit being a controlled displacement of the field-mode. Recently, we have developed an analogous model which employs a d-dimensional qudit ancilla  with a displacement operator defined in the discrete phase space of the qudit [23, 24]. These models have been shown to require a lower number of operations to implement certain gate sequences than a direct implementation of the circuit model [19, 22, 25]. However, neither of these models can implement a universal gate set on the register using only this ancilla-register interaction and so, although no interactions between register qubits are required, some further direct access is needed to the register qubits to implement some basis-changing single-qubit unitary [22, 26].
In order to implement useful quantum computations, it will be necessary to have register qubits with as long a coherence time as possible. However, if it is necessary to access each register qubit to implement multiple forms of control this will potentially introduce many sources of decoherence. Limiting the forms of access required to the register qubits may help to isolate the register more effectively, and so, motivated by this, the measurement-based ancilla-driven quantum computation (ADQC) [27–29] and more recently the globally unitary ancilla-controlled quantum computation (ACQC)  schemes were developed. In both of these models the access to the register qubits is limited to one fixed-time interaction between a single register qubit (at a time) and an ancilla qubit, where the qubits are not necessarily of the same physical type. The additional direct access to the register that was required in the qubus architecture, and analogous qudit model, is replaced by local unitary operations on the ancilla and, in the case of ADQC, ancilla measurements.
Although both the ancilla-driven and ancilla-controlled models require the minimum possible access to the computational register, they replace the local control of the register with local control of the ancillary system and so still require more than one fixed quantum gate to implement universal computation. Halil-Shah and Oi  have recently shown that the measurement-based ancilla-driven model can be adapted so that no local control, beyond ancilla preparation in a fixed state, is required of either the ancilla or the register. In this model, the computation can be achieved using a fixed interaction and ancilla measurement in the computational basis alone. However, this requires a stochastic repeat-until-success style gate scheme , whereby one has to wait until a random walk through the set of unitaries is within the required precision of the desired unitary. In this paper we will show that it is possible to develop deterministic models that require only a single fixed ancilla-register interaction and ancilla preparation in the computational basis with no ancilla measurements necessary. Such schemes require a minimal level of control of both the ancillary and register systems whilst allowing for universal quantum computation. Hence, we will refer to such models as minimal control models of ancilla-mediated quantum computation, and we will often drop the reference to ancilla-mediation for brevity. In Section 3 we introduce two such schemes based on locally inequivalent interactions. The first of these models requires multiple ancillas to implement entangling two-qubit gates on the register and so, although it requires minimal control, it has an overhead in terms of ancilla use. We then develop an alternative minimal control model which does not have this ancilla overhead and requires only three ancilla-register interactions per two-qubit entangling gate, the minimum possible in any unitary scheme . We briefly discuss the physical implementation of these models before concluding in Section 4. We begin in Section 2 with some essential definitions.
for some .
3 Minimal control ancilla-mediated quantum computation
We now present two schemes of ancilla-mediated quantum computation that require only a single fixed-time ancilla-register interaction and ancilla preparation in the computational basis and hence are minimal control models.
3.1 A first minimal control model
We introduce our first model by giving a general form for an ancilla-register interaction which under certain conditions can implement a universal gate set on a register of qubits within the constraints of minimal control. We give an explicit construction for the application of a universal gate set on the register before comparing this model to the measurement-based scheme of Halil-Shah and Oi  and giving a simple example of an interaction that obeys the required constraints.
3.1.1 A general interaction
3.1.2 Discussion and comparison with a measurement-based model
We compare this minimal unitary model to the recently proposed measurement-based minimal scheme of Halil-Shah and Oi . In their model the computation is achieved using only a fixed interaction, ancilla preparation in a fixed state, and measurement in the computational basis. The fixed initial ancilla state is compensated for by the measurement which projects the ancilla onto states in the computational basis. Only one and two ancilla-register interactions are required to implement the single and two-qubit gates respectively. However this model results in a probabilistic repeat-until-success style gate scheme , whereby one has to wait until a random walk through the set of unitaries is within the required precision of the desired unitary. Although the model presented here is entirely deterministic, there is an overhead, that in general may be large, to implement the two-qubit gates. We note that this overhead does not appear if an ancilla measurement is performed after the first two interactions in Figure 2, for an appropriately initialised ancilla and measurement basis, and in this case both models are of a similar form.
We note that if vu is diagonal in the computational basis the additional ancillas are not required, however in this case and will commute and so cannot form a universal set for . In general, the required additional ancillas for the two-qubit gates may create an impractical overhead. However, we now demonstrate that there exists an appropriate form for K that is universal and has a low overhead for additional ancillas for each entangling gate.
3.1.3 A specific interaction
A specific suitable choice for the ancilla-register interaction K is given by taking and . It is straightforward to show that this gives and . We have that and so . It then follows that and form a universal set for as H and T are a universal set for single-qubit unitaries . It is necessary to implement on each register qubit to implement the sequence of Eq. (8) and Figure 2. We have that and so the sequence of Eq. (8) and Figure 2 can be implemented using 14 ancillas prepared in the state and one ‘entangling’ ancilla, that mediates the gate, prepared in any state.
3.2 A second minimal control model
We now present an alternative minimal control model which does not require additional ancillas. As before, we will introduce the model with a general interaction which with certain restrictions can be used to implement a minimal control model and then give a specific example of a simple suitable interaction.
3.2.1 A general interaction
where and we assume that is a universal set for . This is represented in the circuit diagram of Figure 3c. Hence, as we have shown how to implement a two-qubit entangling gate and a universal set for on the register then this is a minimal control model of ancilla-mediated quantum computation.
3.2.2 Discussion and comparison with other models
This model requires three interactions for each entangling two-qubit gate on the register which, although greater than the two needed with the aid of ancilla measurement in ADQC  and the minimal extension of Halil-Shah and Oi , is the minimum possible in any measurement-free scheme . Furthermore, in contrast to the first minimal control model, there is no requirement for multiple ancilla qubits to implement the entangling gates. Finally, we note that the two-qubit gates on the register are implemented in an identical fashion to those in the ACQC model  and L obeys the required conditions to be universal for that model.
3.2.3 A specific interaction
A simple example of a specific form for the interaction such that and form a universal set for (and hence may implement this minimal control model) is given by taking , and . We then have that and hence and . A proof of the universality of for is given in the Appendix. The entangling gate induced on a pair of register qubits from the sequence of Eq. (13) is then and this can easily simulate as in this case .
3.3 The physical implementation of minimal control models
If we consider the second minimal control model and take the fixed ancilla-register gate to be the unitary implemented by applying to the ancilla and register qubit for a time , i.e. , followed by a fixed ancilla rotation of the form , we have that . Hence, this gives and which we have shown to be a universal set for when and so this form for is appropriate for implementing the second minimal control model. With this simple interaction Hamiltonian, , local control of the ancilla is required. However, we see that this is a fixed gate on the ancilla after every ancilla-register interaction via and hence this can be a fixed element in an experimental setup or incorporated into the natural evolution of the ancilla between interactions.2 For example, if the ancillary qubit is photonic the local operation can be performed by fixed linear optics  after each ancilla-register interaction. Indeed, the use of ancillary photons to mediate gates has been demonstrated in many experimental setups, for example with atomic [37, 38] or spin [13, 14] qubits.
Interactions with the form arise naturally in spin systems, with one example of an implementation of given by the coupling between quantum dot resonant exchange qubits . A particularly relevant physical system to ancilla-mediated models is the coupling of nuclear spins via ancillary electronic spins in nitrogen-vacancy (NV) defects in diamond [40–42] and in such setups it may be possible to engineer the Hamiltonian . Although in some physical realisations, such as the photonic case discussed above, the fixed local operation on the ancillary qubit after each interaction is convenient or natural, in others it may be problematic and negate the benefits of the models introduced herein. However, it is also possible to find Hamiltonians that directly implement suitable interactions for either of the models proposed in Sections 3. In certain systems the Hamiltonian is highly tuneable, with an example being those involving superconducting qubits , and due to the long coherence times of ensembles of electron spins in NV centers  a particularly promising physical system for ancilla-mediated quantum computing is arrays of spin ensembles in diamond (each spin ensemble is an effective qubit) coupled by ancillary flux qubits [7–9]. Indeed, coherent coupling in such a system as been demonstrated . It would be interesting to consider which physical systems have Hamiltonians that are naturally suited to generating appropriate interactions for the models introduced herein and we leave a more detailed study of this for future work.
We have presented two unitary models of ancilla-mediated quantum computation that require only minimal control of both the ancillary and register systems. The only control necessary in these models to implement universal quantum computation on a register of qubits is a single fixed-time ancilla-register interaction between one ancilla qubit and one register qubit (at a time) and ancilla preparation in the computational basis. The first of these models is based on maximally entangling interactions that are locally equivalent to CZ and requires multiple ancilla qubits to mediate two-qubit entangling gates on the register. This model is similar in many respects to the minimal measurement-based ancilla-mediated model of Halil-Shah and Oi , in which the requirement for ancilla preparation is replaced with the need for ancilla measurements in the computational basis, but is deterministic rather than stochastic. The second of these models removes the need for multiple ancillas to mediate each entangling gate by employing interactions that utilise the SWAP gate in a similar manner to the model known as ancilla-controlled quantum computation . As in the ancilla-controlled model, only three ancilla-register interactions are required to implement a two-qubit entangling gate on the register, which is the minimum possible in any scheme that does not include measurements , and two for a single-qubit gate. Again, due to the global unitarity of the model the computation is deterministic and is based on a finite gate set composed of one two-qubit entangling gate and two single-qubit gates that form a universal set for single-qubit unitaries. We conjecture that these models require the minimal possible level of control for a unitary ancilla-mediated scheme.
Therefore, for both and we have that and hence φ is an irrational multiple of π  and where . As φ is an irrational multiple of π we can approximate to arbitrary accuracy any rotation around the axis by m applications of , with m a finite integer. As these axes of rotation are not parallel then any arbitrary rotation can be decomposed into rotations around these axes . This then proves that and and hence and are a universal set for .
The two-qubit gate in this first minimal control model can be seen to employ essentially the same geometric methods as the qudit ancilla model we introduced in Ref.  and hence also the qubus model. In the qudit ancilla model the ancilla interacts with the register via displacements of the ancilla (with the displacements defined in the discrete lattice phase space of a qudit) controlled by a register qubit. In this model with a qudit of dimension 2, i.e. a qubit ancilla, controlled displacements are the and gates. The two-qubit gates between register qubits j and k are then mediated by a sequence of the form . This requires two different interactions between the ancilla and register. The model presented here essential uses this gate method but removes the need for two different interactions by including a Hadamard gate on the ancilla in the interaction definition as . It is the additional local gates and (needed to make the gate universal without additional single-qubit gates) that then results in the need for additional ancillas to mediate the two-qubit gates in the first minimal control model. Finally, this gate method can be considered to be geometric as can be considered to create a closed loop in the discrete lattice phase space of a qubit - the details of this are in Ref. .
Note that this is different to ADQC and ACQC in which the required rotations on the ancilla depend on the gate that is to be implemented (and previous measurement outcomes in the case of ADQC).
TJP was supported by a University of Leeds Research Scholarship.
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