21
One‐Way Quantum Computation

Dan Browne1 and Hans Briegel2

1 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK

2 Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, 6020 Innsbruck, Austria

21.1 Introduction

The circuit model of quantum computation (1–3) has been a powerful tool for the development of quantum computation, acting as both a framework for theoretical investigations and a guide for experiment. In the circuit model (also called the network model), unitary operations are represented by a network of elementary quantum gates such as the CNOT gate and single‐qubit rotations. Many proposals for implementing quantum computation have been designed around this model, including physical prescriptions for implementing the elementary gates. By formulating quantum computation in a different way, one can gain both a new framework for experiments and new theoretical insights. One‐way quantum computation (4) has achieved both of these.

Measurements on entangled states play a key role in many quantum information protocols, such as quantum teleportation and entanglement‐based quantum key distribution. In these applications, an entangled state is required, which must be generated beforehand. Then, during the protocol, measurements are made, which convert the quantum correlations into, for example, a secret key. To repeat the protocol, a fresh entangled state must be prepared. In this sense, the entangled state, or the quantum correlations embodied by the state, can be considered a resource, which is “used up” in the protocol.

In one‐way quantum computation, the quantum correlations in an entangled state, called a cluster state (5) or graph state (6), are exploited to allow universal quantum computation through single‐qubit measurements alone. This computational model is depicted in Figure 21.1. The quantum algorithm is specified in the choice of bases for these measurements and the “structure” of the entanglement (as explained below) of the resource state. The name “one‐way” reflects the resource nature of the graph state. The state can be used only once, and (irreversible) projective measurements drive the computation forward, in contrast to the reversibility of every gate in the standard network model.

This chapter provides an introduction to one‐way quantum computation, and several of the techniques one can use to describe it. In this section, we introduce graph and cluster states and develop a notation for general single‐qubit measurements. In Section 21.2, we introduce the key concepts of one‐way quantum computation with some simple examples. After this, in Section 21.3, we shall investigate how one‐way quantum computation can be described without using the quantum circuit model. To this end, we shall introduce a number of important tools including the stabilizer formalism, the logical Heisenberg picture, and a representation of unitary operations especially well suited to the one‐way quantum computation model. In Section 21.4, we briefly describe many proposals for implementing one‐way quantum computation in the laboratory. In Section 21.5, we conclude with a brief survey of some recent research developments in measurement‐based quantum computation updated in January 2017 for the second edition of this book.

A different perspective of one‐way quantum computation and measurement‐based computation in general can be found in the following reviews (7,8). A comprehensive tutorial and review on the properties of graph states can be found in (9).

21.1.1 Cluster States and Graph States

Cluster states and graph states can be defined constructively in the following way ( 5, 9). With each state, we associate a graph, a set of vertices and edges connecting vertex pairs. Each vertex on the graph corresponds to a qubit. The corresponding “graph state” may be generated by preparing every qubit in the state images and applying a controlled images (CZ) operation images on every pair of qubits whose vertices are connected by a graph edge. Cluster states are a subclass of graph states, whose underlying graph is an images ‐dimensional square grid. The extra flexibility in the entanglement structure of graph states means that they often require far fewer qubits to implement the same one‐way quantum computation. However, there are a number of physical implementations where the regular layout of cluster states means that they can be generated very efficiently (see Section 21.4 ).

Scheme for One-way quantum computation consists of single-qubit measurements in certain bases and in a certain order on an entangled resource state.

Figure 21.1 One‐way quantum computation consists of single‐qubit measurements in certain bases and in a certain order on an entangled resource state. Cluster states have a square‐lattice structure (a) while the freedom of choosing specific general graph states such as illustrated in (b) can reduce the number of qubits needed for a given computation significantly.

21.1.2 Single‐Qubit Measurements and Rotations

Single‐qubit measurements in a variety of bases play a key role in one‐way quantum computation, so here we introduce a convenient and compact way to describe them. Using a Bloch sphere picture, every projective single‐qubit measurement can be associated with a unit vector on the sphere, which corresponds to the images eigenstate of the measurement. We can then parameterize observables by the colatitude images and longitude images of this vector (illustrated in Figure 21.2). We shall write this compactly as a pair of angles images .

Unitary operations corresponding to rotations on the Bloch sphere have the following form. A rotation around the images ‐axis (where images is images , images , or images ) by angle images can be written

21.1 equation

For brevity and clarity, we use the notation images , and so on in the rest of this chapter. We also adopt standard notation for the eigenstates of images and images :

21.2 equation

A measurement with angles images corresponds to a measurement of the observable images . One way of implementing such a measurement is to apply the single‐qubit unitary images to the qubit before measuring it in the computational basis.

Image described by caption and surrounding text.

Figure 21.2 Single‐qubit projective measurements will be represented by the pair of angles images of the colatitude images and longitude images of their images eigenstate on the Bloch sphere. This corresponds to a measurement of the observable images .

21.2 Simple Examples

Many of the features of one‐way quantum computation can be illustrated in a simple two‐qubit example. Consider the following simple protocol; a qubit is prepared in an unknown state images . A second qubit is prepared in the state images . A CZ operation is applied on the two qubits. The state of the qubits is then

21.3 equation

The first qubit is now measured in the basis images , where images is a real parameter. Using the notation introduced in Section 21.1.2, this measurement is denoted images . This corresponds, in the Bloch sphere picture, to a unit vector in the images –images plane at angle images to the images ‐axis. There are two possible outcomes to the measurement, which occur with equal probability. If the measurement returns the images eigenvalue, the second qubit will be projected into the state

21.4 equation

If the images eigenvalue is found, the state of qubit two becomes

21.5 equation

We can represent both possibilities in a compact way if we introduce the binary digit images to represent a measurement outcome of images . The state of qubit two can then be written, up to a global phase,

21.6 equation

We see that the unknown input state, which was prepared on the first qubit, has been transferred to qubit two without any loss of coherence. In addition to this, it has undergone a unitary transformation: images . Notice that the angle of the rotation images is set in the choice of measurement basis. The unitary transformation images is accompanied by an additional Pauli transformation (images ) when the measurement outcome is images . This is a typical feature of one‐way quantum computation; due to the randomness of the measurement outcomes, any desired unitary can be implemented only up to random but known Pauli transformations. Since these Pauli operators are an undesired by‐product of implementing the unitary in the one‐way model, we call them “by‐product operators” ( 4,10). As we shall see below, these extra Pauli operations can be accounted for by altering the basis of later measurements, making the scheme deterministic but introducing an unavoidable time‐ordering.

Image described by caption and surrounding text.

Figure 21.3 The one‐way graph and measurement patterns for (a) the single‐qubit operation images and (b) an arbitrary single‐qubit operation, images , when the measurement angles are set images , images and images , and images is the binary measurement outcome of the measurement on qubit images . Note that this imposes an ordering in the measurements of this pattern. This second pattern is made by composing three copies of the pattern (a) with differing measurement angles as described in the text. Pattern (c) implements a CZ operation. Input and output qubits coincide for this pattern.

In Figure 21.3a, this protocol is represented using a graphical notation that we will use throughout this chapter. The input qubit is represented by a square, and the output qubit by a lozenge, a smaller square tilted at images . The CZ operation applied to the two qubits is represented by a line between them. This is an example of a one‐way graph and measurement pattern, or “one‐way pattern” for short, a convenient representation specifying both the entanglement graph for the resource state and the measurements required to implement a unitary operation (always up to a known but random Pauli transformation) in the one‐way model.

So far, the above‐described protocol seems rather different from the description of one‐way quantum computation as a series of measurements on a special entangled resource state. We shall see how the two pictures are related in the following text. First, however, we show how one‐way patterns may be connected together to perform consecutive operations.

21.2.1 Connecting One‐Way Patterns – Arbitrary Single‐Qubit Operations

Due to Euler's rotation theorem, any single‐qubit SU(2) rotation can be decomposed as a product of three rotations images . Thus, by repeating the simple two‐qubit protocol thrice, any arbitrary single‐qubit rotation may be obtained (up to an extra Hadamard, which can be accounted for). Two one‐way patterns are combined as one would expect, the output qubit(s) of one pattern become the input qubit(s) of the next. The main issue in combining patterns is to track the effect of the Pauli by‐product operators, which have accumulated due to the previous measurements.

Concatenating the two‐qubit protocol thrice, with different angles images , images , and images , gives the one‐way pattern illustrated in Figure 21.3b. To see the effect of the by‐product operators from each measurement, let us label the binary outcome from each images . The unitary implemented by the combined pattern is therefore

21.7 equation

Since images and images this can be rewritten

21.8 equation

We can rewrite this further using the identities images and images ,

21.9 equation

Now we have split up the operation in the same manner as that of the two‐qubit example, a unitary plus a known Pauli correction. In this case, however, this unitary is not deterministic – the sign of two of the rotations depends on two of the measurements. Nevertheless, if we perform the measurements sequentially and choose measurement angles images , images , and images , we obtain deterministically the desired single‐qubit unitary.

The dependency of measurement bases on the outcome of previous measurements is a generic feature of one‐way quantum computation, occurring for all but a special class of operations, the Clifford group (described in what follows). This dependency means that there is a minimum number of time steps in which any one‐way quantum computation can be implemented, as discussed further in Section 21.3 .

The Pauli corrections remaining at the end of the implemented one‐way quantum computation are unimportant and never need to be physically applied; they can always be accounted for in the interpretation of the final measurement outcome. For example, if the final state is to be read out in the computational basis any extra images operations commute with the measurements and have no effect on their outcome. Any images operations simply flip the measurement result and thus can be corrected via classical postprocessing.

21.2.2 Graph States as a Resource

It is worth discussing how the above description of one‐way patterns relates to the description of one‐way quantum computation in the introduction, namely as measurements on an entangled resource state. The first observation is that, given a one‐way pattern, all of the measurements can be made after all the CZ operations have been implemented. Second, quantum algorithms usually begin by initializing qubits to a fiducial starting state. This state is usually images on each qubit, but the state images would be equally suitable. When the input qubits of a one‐way graph measurement pattern are prepared in images , then the entangled state generated by the CZ gates is a graph state. Thus, the graph state can be considered a resource for this quantum computation. We shall see in Section 21.4 that for certain implementations, such as in linear optics, the resource description is especially apt.

21.2.3 Two‐Qubit Gates

So far we have seen how an arbitrary single‐qubit operation could be achieved in one‐way quantum computation in a simple linear one‐way pattern. However, for universal quantum computation, entangling two‐qubit gates are necessary. One such gate is a CZ gate. The CZ can be simply implemented within the one‐way framework by using the CZ represented by a single graph‐state edge. This leads to the one‐way pattern illustrated in Figure 21.3c. Notice that here the input qubits are also the output qubits. This is indicated by the superimposed squares and lozenges.

21.2.4 Cluster‐State Quantum Computing

In many proposed implementations of one‐way quantum computation (see Section 21.4 ), square‐lattice cluster states can be generated efficiently, and arbitrarily connected graph states are hard to make. The simple method outlined above for the construction of one‐way patterns will usually lead to graph‐state layouts that do not have a square‐lattice structure. Nevertheless, a cluster state on a large enough square lattice of two or more dimensions is still sufficient to implement any unitary (4). A number of measurement patterns for quantum gates designed specifically for two‐dimensional square‐lattice cluster states can be found in (10).

21.3 Beyond Quantum Circuit Simulation

We have shown that the one‐way quantum computer can implement deterministically a universal set of gates and thus any quantum computation. However, part of the power of one‐way quantum computation derives from the fact that unitary operations can be implemented more compactly than a naive network construction would suggest (11). In fact, we shall see in the following sections that other ways of decomposing unitary operations are more natural and useful. The main tool we shall use in our investigation of these properties is the stabilizer formalism.

21.3.1 Stabilizer Formalism

The stabilizer formalism (12,13) is a powerful tool for understanding the properties of graph states and one‐way quantum computation. Stabilizer formalism is a framework whereby states and subspaces over multiple qubits are described and characterized in a compact way in terms of operators under which they are invariant. In standard quantum mechanics, one uses complete sets of commuting observables in a similar manner, such as in the description of atomic states by “quantum numbers” (see e.g., (14)).

An operator images stabilizes a subspace images when, for all states images ,

21.10 equation

In other words, images is an eigenstate of images with eigenvalue images .

In the stabilizer formalism, one focuses on operators that, in addition to this stabilizing property, are Hermitian members of the Pauli group, that is, tensor products of Pauli and identity operators. The key principle of the stabilizer formalism is to identify a set of such stabilizing operators, which uniquely defines a given state or subspace – that is, there is no state outside the subspace (for a specified system) which the same set of operators also jointly stabilizes. The subspaces (and states) that can be defined uniquely using stabilizing operators from the Pauli group are called stabilizer sub‐spaces (or stabilizer states).

Stabilizer states and subspaces occur widely in quantum information science and include Bell states, GHZ states, many error‐correcting codes, and, of course, graph states and cluster states. Note that there are other joint eigenstates of the stabilizing operators with some images eigenvalues. However, only states with images eigenvalue are “stabilized,” by definition. This set of operators then embodies all the properties of the state and can allow an easier analysis, for example, of how the state transforms under measurement and unitary evolution. Since the product of two stabilizing operators is itself stabilizing, the set of operators that stabilize a subspace has a group structure. It is called the stabilizer group or simply the stabilizer of the subspace. The group can be compactly expressed by identifying a set of generators. For a images ‐qubit subspace in an images ‐qubit system, images generators are required (see Exercise 2).

We do not have enough space here for a detailed introduction to all of the techniques of stabilizer formalism – excellent introductions can be found in ( 3, 12) – but instead we will focus on useful techniques for understanding one‐way quantum computation. Most will be stated without proof but can be verified using the properties of Pauli‐group operators described in (3).

A simple example of a stabilizer state is the state images . Its stabilizer group is generated by images alone. The stabilizer for the tensor product state images is then generated by images operators images acting on each qubit images . From this, we can derive the stabilizer generators for graph states. Consider a stabilizer state transformed by the unitary transformation images . The stabilizers of the transformed state are then given by images . Since the CZ gate transforms images to images under conjugation, we find that the stabilizer generators for graph states have the form

21.11 equation

for every qubit images in the graph. images is the neighborhood of images , that is, the set of qubits sharing edges with images on the graph (this corresponds to nearest neighbors in a cluster state).

21.3.2 A Logical Heisenberg Picture

We are going to use the stabilizer formalism to understand the one‐way patterns that implement unitary transformations in the one‐way model. We shall see that it is convenient to describe logical action of a one‐way pattern in a logical Heisenberg picture (13).

The Schrödinger picture is the most common approach for describing the time evolution of quantum systems. Temporal changes in the system are reflected in changes in the state vector or density matrix, for example, for unitary evolution images or images . The observables that characterize measurable quantities, such as Pauli observables images , images , and images , remain invariant in time. In the Heisenberg picture, on the other hand, time evolution is carried exclusively by physical observables, which evolve images . States and density matrices remain constant in time.

A logical Heisenberg picture, also called a “Heisenberg representation of quantum computation” (13), is a middle‐way between these two approaches, containing elements of both. We shall introduce it with an example, starting in the Schrödinger picture with a single‐qubit density matrix images evolving in time. Since the images ‐qubit Pauli‐group operators form a basis in the vector space of images ‐qubit Hermitian operators, we can write images at time images as

21.12 equation

where images , images , images , and images are real parameters that define the state.

At time images , the state has been transformed through unitary images . In the usual Schrödinger picture, one would reflect this in a transformation of the matrix elements of the state, or, equivalently, of the parameters images , images , images , and images to images , images , and so on. However, one can also write

21.13 equation

By introducing time‐evolving observables images and similar expressions for images and images , we can express this as

21.14 equation

The time evolution is thus captured by the evolution of these logical observables, and the parameters images , images , images , and images remain fixed. Since images , images , and so on define the logical basis in which images is expressed, we call them logical observables.

Since images , determining images and images specifies the evolution images completely. More generally, an images ‐qubit unitary is defined in this picture by the evolution of images and images for each qubit images . It is important to emphasize that the logical observables images , images , and so forth, are no longer equal to the physical observables images , images , and so on, which remain constant in time. Here, time evolution is characterized by the evolution of logical observables. In analogy to the (standard) Heisenberg picture, where physical observables evolve in time, we call this a logical Heisenberg picture.01

The logical Heisenberg picture can be illustrated with some simple examples. First, let us consider a Hadamard images . This is represented in the logical Heisenberg picture through images and images . Second, let us look at the representation of the SWAP gate in this picture. We find that images and images (similarly for the images variables). The logical Heisenberg picture clearly encapsulates the action of these gates; in the case of the Hadamard, we see images and images interchanged and for SWAP, the operators on the two qubits are switched round. In the one‐way quantum computer, logical time evolution is discrete and driven by single‐qubit measurements, so in the following we will often suppress the time labeling images .

A logical Heisenberg picture becomes particularly useful when describing the encoding of quantum information. As well as density matrices, one can represent the evolution of pure state vectors in a logical Heisenberg picture. The time evolution is carried by the logical basis states, the joint eigenstates of images with phase relations fixed by images . Consider a state images imagine we encode it via some unitary transformation images . We would write images , where images and images are the new (encoded) logical basis states. Thus, “encoding” implicitly adopts a logical Heisenberg picture. The state coefficients remain constant while logical basis vectors are transformed.

21.3.3 Dynamical Variables on a Stabilizer Subspace

This formalism can be combined with the stabilizer formalism to track the evolution of logical observables on a subspace of a larger system. The stabilizer group then defines the logical subspace, and the dynamical logical operators track the evolution of this subspace. The logical operators act only to map states around the subspace, therefore they must commute with the stabilizers of that subspace.

Let us use the well‐known three‐qubit error‐correcting code as an example. In this code, the logical images is represented by images and images by images . The stabilizer group for this subspace is generated by images and images . The logical observables associated with this basis are images and images . One can easily verify that these operators have the desired action on the logical basis states.

However, even though the logical basis is entirely symmetric under interchange of the qubits, the logical images is not. Due to the symmetry of the situation, one would expect that images and images would be equivalent to the physical representation of images we have chosen above. That these operators have the same action on the logical basis states is easy to confirm, and it reflects an important characteristic of logical operators on a subspace, namely that they are not unique. Given a stabilizer operator for the subspace images and logical operator images , the product images has the same action on the logical subspace as images . Thus, there are a number of physical representations for a given logical observable. Formally, this set is in fact a coset of the stabilizer group. In order to define this set, only one member of the set need be specified. When we write a particular physical operator corresponding to images , this is just a “representative” of the whole coset.

21.3.4 One‐Way Patterns in the Stabilizer Formalism

We introduced the term “one‐way pattern” to describe a layout of qubits, graph‐state edges, and measurements, which implements a given unitary in the one‐way model. More specifically, the patterns contain a set of qubits labeled input qubits and a set of labeled output qubits, a set of auxiliary qubits, and a set of edges connecting those qubits. We will now show how using the rules for transforming stabilizer subspaces under measurement, the one‐way pattern will lead to the transformation of the logical operators images and images . This is a logical Heisenberg picture representation of the desired unitary images , plus the displacement of the logical state from input qubit(s) images to output qubit(s) images . The extra factor images reflects the presence of by‐product operators (due to the randomness of the measurement outcomes) since images and images .

21.3.5 Pauli Measurements

Scheme for Any n-qubit Clifford-group operation implemented by a one-way pattern with 2n-qubits.

Figure 21.4 Any images ‐qubit Clifford‐group operation may be implemented (up to local Clifford corrections) by a one‐way pattern with images ‐qubits. Dotted lines represent possible edges in the patterns.

Before we consider one‐way patterns with general one‐qubit measurements, let us first consider patterns consisting solely of Pauli measurements. These measurements change the logical variables' encoding according to the desired evolution of the logical state. As the logical evolution is unitary, each measurement must reveal no information about the logical state. By considering commutation relations, one can show that these requirements are equivalent to demanding that the measured observable anticommute with at least one stabilizer generator.

The effect of performing a measurement of a (multiqubit) Pauli observable images on a subspace is as follows (such methods are described in more detail in (3)). If images does not commute with the complete stabilizer group, one can always construct a set of stabilizer generators such that only one of the generators images anticommutes with images . The stabilizers that commute with images must also stabilize the transformed subspace after the measurement, which will be an eigenspace of images with eigenvalue images . Thus, images will itself belong to the new stabilizer. We can thus construct a set of generators for the stabilizer of the transformed subspace, by simply replacing images , which anticommutes with images , with images .

The logical observables transform in a similar way. This time, just one member of the coset for each logical observable needs to be found, which commutes with images . If the representative logical operator images commutes with images , it remains a valid representative logical operator after the measurement (the full coset will be different though due to the changed stabilizer). If images does not commute with images , then the product images does commute, so logical operators for the transformed subspace are easy to find.

A final step involves finding a reduced description of the state, which now ignores the unimportant measured qubit. This is achieved by choosing a set of stabilizer generators where all but one (images itself) act as the identity on the measured qubit. This is achieved by multiplying all the generators not already in this form with images . In the same way, representative logical operators can be chosen that are also restricted to the unmeasured qubits.

After all but the designated output qubits in a pattern have been measured, the one‐way pattern has been completed. The reduced description of the output qubits has a stabilizer group consisting of the identity operator alone and logical operators have become images and images . We interpret this in the logical Heisenberg picture. The one‐way pattern has implemented the unitary transformation images plus known Pauli corrections, and the logical subspace has been physically displaced from the input qubits images to output qubits images .

Illustration of full orbit of locally equivalent four-qubit graph states.

Figure 21.5 The full orbit of locally equivalent four‐qubit graph states. Each graph state is obtained from the previous one by applying the “local complementation rule.” ( )

Hein et al. 2004 (6). Copyright 2004, American Physical Society.

This method can be used to design and verify one‐way patterns (e.g., see Exercise 3). It may seem complicated for such simple examples, but its power lies in its generality. In the next section, we show how measurement patterns for arbitrary Clifford operations may be evaluated using these techniques.

21.3.6 Pauli Measurements and the Clifford Group

In the previous section, all of the transformations of the logical observables keep their physical representations within the Pauli group. Unitary operators that map Pauli‐group operators to the Pauli group under conjugation are known as Clifford‐group operations. The Clifford group is the group generated by the CZ, Hadamard, and images gates. Since all of these gates can be implemented by one‐way patterns with Pauli measurements only (i.e., by choosing images or images in Figure 21.3a) any Clifford‐group operation can be achieved by Pauli measurements alone.

The Clifford group plays an important role in quantum computation theory. Clifford‐group circuits are the basis for most quantum error correction schemes, and many interesting entangled states (including, of course, graph states) can be generated via Clifford‐group operations alone. However, Gottesman and Knill showed that notwithstanding this, Clifford‐group circuits acting on stabilizer states (such as the standard input images ) can be simulated efficiently on a classical computer (15,16). This is because of the simple way the logical observables transform (in the logical Heisenberg picture) under such operations.

Let us consider the effect of the by‐product Pauli operators, generated every time a measurement outcome is images , when Clifford operations are implemented in the one‐way quantum computer. Given a Clifford operation images , by the definition of the Clifford group, images where images and images are Pauli‐group operators. Therefore, images meaning that interchanging the order of Clifford operators and Pauli corrections will leave the Clifford operation unchanged. This means that there is no need to choose measurement bases adaptively. We thus see that in any one‐way quantum computation all Pauli measurements can be made simultaneously in the first measurement round.

These results imply that Pauli measurements on stabilizer states will always leave behind a stabilizer state on the unmeasured qubits. Additionally, any stabilizer state can be transformed to a graph state by local Clifford operations (17,18). Furthermore, this graph state is in general not unique, by further local Clifford operations a whole family of locally equivalent graph states can be achieved ( 6, 18). The rules for this local equivalence are simple – a graph can be transformed into another locally equivalent graph by “local complementation” (18), which is a graph‐theoretical primitive (19). In local complementation, a particular vertex of the graph is singled out and the subgraph given by all vertices connected to it is “complemented” (i.e., all present edges are removed and any missing edges are created). The set of locally equivalent four‐qubit graph states is illustrated in Figure 21.5.

This theorem allows us to understand the effect of Pauli measurements on a graph state in a new way. Any Pauli measurement on a graph state simply transforms it (up to a local Clifford correction) into another graph state. How the graph is transformed and which local corrections must be applied are graphically described in ( 6,20). The rule for images ‐measurements is simple, and the measured qubit and all edges connected to it are removed from the graph. If the images eigenvalue was measured, extra images transformations on the adjacent qubits must be applied to bring the state to graph‐state form. Rules for images and images measurements are more complicated and can be found in (6).

Since the effect of Pauli measurements is to just transform the graph, given any one‐way pattern containing Pauli measurements, the transformation rules can be used to find a one‐way pattern, which implements the same operation with fewer qubits. The local corrections can often be incorporated in the bases of remaining measurements. If not they lead to an additional local Clifford transformation on the output qubit. Since the Pauli measurements correspond to the implementation of Clifford‐group operations, this leads to a stronger result than the Gottesman–Knill theorem. All Clifford operations wherever they occur in the quantum computation are reduced to classical preprocessing of the one‐way pattern. A further consequence is that any images ‐qubit Clifford‐group operation can be implemented (up to the local Clifford corrections) on a images ‐qubit pattern, as illustrated in Figure 21.4.

21.3.7 Non‐Pauli Measurements

The method above does not yet allow us to treat non‐Pauli measurements, specified by measurement directions other than along the images ‐, images ‐, or images ‐axis. However, one can still treat such measurements within the stabilizer formalism. The stabilizer eigenvalue equations (Eq. 21.10) can be rearranged to generate a family of non‐Pauli unitary operations, which also stabilize the subspace (10). Consider the state images stabilized by operator images . We rearrange the stabilizer equation as follows:

21.15 equation

thus for all images ,

21.16 equation

Thus, we have a unitary images , which itself stabilizes images . This implies that

21.17 equation

Similar unitaries and similar expressions can be generated from any stabilizer operator. We show in the next section how this technique allows a simple analysis of the one‐way pattern for general unitaries diagonal in the computational basis, and, in fact, the technique allows one to understand any one‐way pattern solely within the stabilizer formalism and was used to design and verify many of the gate patterns presented in (10). This indicates that the effect of non‐Pauli measurements in a one‐way quantum computation can always be understood as the implementation of a generalized rotation images , where images can be any images ‐qubit Pauli‐group operator. We shall discuss the consequences of this further in section 21.3.9.

21.3.8 Diagonal Unitaries

Earlier in the chapter we saw that a CZ gate can be implemented such that the input qubit is also the output qubit. Coinciding input and output qubits in a one‐way pattern reduces the pattern size so it is natural to ask which unitaries can be implemented this way, and one can show (see Exercise 4) that it is only those unitaries diagonal in the computational basis. In fact, there is a simple one‐way pattern for any diagonal unitary transformation. Any such images ‐qubit operator can be written (up to a global phase) in the following form:

21.18 equation

where images is equal to the identity if images and images acting on qubit images when images , and the sum is over all binary vectors images of length images .

Each element of this product is a generalized rotation acting on a subset of the qubits and has a very simple implementation in the one‐way quantum computer. To illustrate this, consider the two‐qubit transformation images . This can be implemented on a one‐way pattern with three qubits as illustrated in Figure 21.6. In this pattern, the qubits labeled 1 and 2 are the joint input–output qubits, and qubit images is an ancilla. The entanglement graph has two edges connecting images to 1 and 2. A measurement in basis images , that is, of observable images , implements images on the input state, with by‐product operator images .

To see this, we recall that the stabilizer for the subspace corresponding to such a graph is images . The corresponding eigenvalue equation can be transformed, as described in the previous section, to generate the stabilizing unitary images . Measuring qubit images in basis images is equivalent to performing images and then measuring images ; hence, the one‐way pattern implements the logical unitary images .

Image described by caption and surrounding text.

Figure 21.6 The one‐way pattern that implements the unitary “double‐z rotation” images . Note that input and output qubits coincide.

Illustration of Arbitrary diagonal unitaries may be implemented in a single round of measurements by measurement patterns with coinciding input and output qubits.

Figure 21.7 Arbitrary diagonal unitaries may be implemented in a single round of measurements by measurement patterns with coinciding input and output qubits. This example shows an arbitrary diagonal three‐qubit unitary images . For example, by setting the angles to images and images , we obtain a control–control Z gate or “Toffoli‐Z gate.” See (10) for a cluster‐state implementation of this gate.

We can generalize this pattern to quite general images ‐qubit diagonal unitaries. (Verify this in Exercise 5.) For example, the pattern for an arbitrary diagonal three‐qubit unitary is given in Figure 21.7. This is a highly parallelized and efficient implementation of the unitary.02

Since the by‐product operators for these patterns are diagonal themselves, they commute with the desired logical diagonal unitaries. Thus, there is no dependency in the measurement bases on the outcome of measurements within this pattern and all measurements can be achieved in a single measurement round. Thus, not only can a quantum circuit consisting of Clifford gates alone be implemented in a single time step – this is true for any diagonal unitary followed by a Clifford network.

21.3.9 Gate Patterns Beyond the Standard Network Model – CD‐Decomposition

We have seen that one can construct one‐way patterns to implement a unitary operation described by a quantum circuit by simply connecting patterns together for the constituent gates. Furthermore, such patterns can be made more compact by evaluating the graph‐state transformations corresponding to any Pauli measurements present. This can change the structure of the pattern such that the original circuit is hard to recognize (see e.g., the quantum Fourier transform patterns in (6)).

We have also seen that non‐Pauli measurements in a measurement pattern lead to generalized rotations on the logical state of the form images , where images is some Pauli‐group operator. The implementation of any non‐Clifford unitary on the one‐way quantum computer is thus best understood as a sequence of operators of this form. Two such operators images and images may, if images be combined to give images . In general, any operators of the form images , where images , can be diagonalized by a Clifford‐group element images to images , where images is a diagonal unitary. Composing two operations this form, for example, images and images will give images , where images , and we call the casting of a unitary in this form a CD‐decomposition.

There are several observations to be made about such decompositions. We have already seen that both diagonal unitaries and Clifford‐group operations have compact implementations in one‐way quantum computations. This means that CD decompositions are very useful in the design of compact one‐way patterns. One simply combines the one‐way patterns for diagonal unitaries presented above with patterns for Clifford operations, which we have seen require at most images qubits for an images ‐qubit operation and which can be constructed either by employing Pauli transformation rules on a pattern for a network of CZ, Hadamard and images gates, or by inspecting the logical Heisenberg form of the operation. In (10) this decomposition, together with stabilizer techniques described earlier, was used to design cluster‐state implementations for several gates and simple algorithms including controlled Z‐rotations and the quantum Fourier transform (QFT).

A further advantage in working with a CD decomposition is that it immediately provides an upper bound in the number of time steps needed for the implementation of the one‐way pattern. This is simply the number of “CD units” in the decomposition. We saw in section 21.3.8 that a single CD unit can be implemented in a single time step. Each CD unit, in turn, will create by‐product operators, which may need to be accounted for in the choice of measurement bases for the following diagonal unitaries. A decomposition that minimized the number of CD units would give a (possibly tight) upper bound on the minimal number of time steps and would be one measure of how hard the unitary is to implement in the one‐way model. For example, Euler's rotation theorem tells us that the optimal CD decomposition for an arbitrary rotation consists of three CD units and correspondingly requires three measurement rounds for implementation on the one‐way quantum computer.

Note that there is considerable freedom in choosing a CD form. For example, one can construct the decomposition such that all the diagonal gates are solely local, single‐qubit operations and only the Clifford gates are nonlocal. This gives a degree of flexibility in the design of one‐way patterns.

Quantum circuits described in terms of Clifford‐group gates plus rotations can readily be cast in CD form by decomposing the rotations into images ‐axis rotations and Hadamards. One can then reduce the size of the corresponding pattern by applying the Pauli measurement transformation rules03 . Quantum circuits for the simulation of general Hamiltonians are usually expressed using the Trotter formula (see (3)), which leads to unitaries which are a sequence of generalized rotations, which can be cast in CD form in a straightforward manner. Thus, the one‐way quantum computer is very well suited to Hamiltonian simulation (see e.g., (22)), which will be an important application of quantum computers.

21.4 Implementations

21.4.1 Optical Lattices

Beyond its theoretical value, there are a number of physical implementations where one‐way quantum computation gives distinct practical advantages. One of these is in systems where graph states or cluster states can be generated efficiently, such as “optical lattices.” In an optical lattice, cold neutral atoms are trapped in a lattice structure, given by the periodic potential due to a set of superposed laser fields. The potential “seen” by each atom depends on its internal state. This means that neighboring atoms in different states can be brought close together by changing the relative positions of the minima of the periodic potentials, leaving an interaction phase on the atoms' state (23). If this is timed such that this interaction phase is images the process implements, essentially, a CZ gate between the two atoms. However, every atom in the lattice will be affected when these potentials move and thus CZ gates can be implemented between neighboring qubits across the lattice simultaneously. Thus, by preparing all atoms in a superposition of these internal states beforehand, a very large cluster state can be generated very efficiently. There has been much progress in the generation and manipulation of ultracold atoms in optical lattices in the laboratory (24), and a number of schemes for the generation of arbitrary graph states in these systems have been proposed (25). Possibly, the most difficult obstacle to overcome for the implementation of one‐way quantum computation in optical lattices is the difficulty in addressing individual atoms in the lattice.

21.4.2 Linear Optics and Cavity QED

Photons make excellent carriers on quantum information and are relatively decoherence free. A key difficulty in implementing universal quantum computation using photons is that two‐qubit gates such as CZ cannot be implemented by the simple linear optical elements of the optics laboratory (e.g., beam splitters and phase shifters) alone. By employing photon number measurements, nondeterministic entangling gates are possible. Most times, however, the gate fails, and this failure leads to the measurement of the qubits' state, which disrupts the computation. Naively, one would expect that scaling this up into a circuit would lead to an exponential decrease in success probability, but, by using a combination of techniques including gate teleportation (26) and error correction, scalable quantum computation is indeed possible (27). A key disadvantage of this particular approach, however, is that each gate requires a large number of ancilla photons in a difficult‐to‐prepare entangled state.

A much more efficient strategy is to use the nondeterministic gates to build an entangled resource state for measurement‐based quantum computation (28,29). Cluster states can be generated efficiently (30) using so‐called “fusion operations,” which can be performed (nondeterministically) with simple linear optics. Fusion operations ( 30,31) are implementations of operators such as images , which, when applied to two qubits in different graph states, replace both qubits by a single one which inherits all the graph‐state edges of each, thus “fusing” the two graph states together. Three‐ and four‐qubit graph states have been created in the laboratory using methods based on downconversion and postselection (32) and fusion measurements (33). Single‐qubit measurements on these states demonstrated many of the key elements of one‐way quantum computation (32). More details of linear optical quantum computation can be found in other chapters of this book and in a comprehensive review article (34).

Quantum computation with photons is not the only scenario where gates are inherently nondeterministic. Similar techniques can be used to implement nondeterministic gates between atoms or ions trapped in separate cavities. Cavity QED implementations of the one‐way quantum computer is a fast‐developing area, and there have been a number of promising experimental proposals (35).

21.5 Recent Developments

We have updated this section for the second edition of this book in 2017 to cover developments since the first edition appeared in 2006. During this time there has been a wealth of research related to measurement‐based quantum computation (e.g., Ref. (4) has been cited over 2000 times). We therefore cannot be exhaustive here, but we hope to reflect the diversity of this research.

There have been a number of proposals of generalizations and variants of one‐way quantum computation. Applying the powerful framework of tensor networks, Gross and Eisert (36) proposed a general framework for quantum computation that did not rely on cluster states or the stabilizer formalism, but admitted a much broader family of entangled states, which were resources for universal quantum computation. The entanglement properties a family of states must satisfy to be universal resources was the focus of vibrant debate and discussion (37). Indeed, it was even shown that in some circumstances states could have too much entanglement to be useful computational resources (38).

One‐way quantum computation was generalized beyond qubits to qudits (39) and continuous‐variable systems (40,41). The latter case provided an entirely new avenue toward experimental realization of one‐way quantum computation, which we will return to below. A further variant of one‐way quantum computing is ancilla‐driven quantum computing (42). Ancilla‐driven quantum computing adapts some of the concepts of one‐way quantum computing combining measurement on ancilla qubits with a constant coupling with other qubits.

Finally, the notion of graph states (9) have been generalized to hypergraph states (43). A hypergraph generalizes the notion of a graph. In a graph, pairs of vertices are connected by edges. In a hypergraph, sets of more than two vertices can be connected via so-called hyperedges. An entangled state can be analogously defined, replacing the CZ gates of graph edges with multicontrolled Z gates for hyperedges. Hypergraph states inherit a number of the properties of graph states but also have a rich set of their own properties.

A symbolic syntax for one‐way quantum computation, the measurement calculus, was developed by Danos et al. (44), which provided a rich and robust way to study and reason about one‐way quantum computations, which led to progress in understanding determinism in one‐way quantum computations not derived from the circuit model (45), among a number of other advances.

One‐way quantum computing has remained important for fault‐tolerant quantum computing, and the leading architectures for fault‐tolerant quantum computation use a combination of transversal unitary gates (circuit model) and code deformation (a form of measurement‐based quantum computation, where by changing the measurements used to detect errors encoded quantum data undergo logical gates) together with state injection (related to gate‐teleportation). In particular, topological code deformation for surface codes, the basis of experimental architectures (such as the one pursued by John Martinis at Google (46)) was derived from a cluster‐state fault‐tolerance approach (47,48).

One‐way quantum computing has produced new perspectives for condensed matter. The property of being a computational resource state is a property that one can look for in interacting many‐body quantum systems and in the ground state of their Hamiltonians (49), and one can consider phase transitions in computational resource power (50). Symmetry‐protected topological order, a phenomenon studied in condensed matter physics, was shown to be closely related to one‐way quantum computation and each field provided new insights for the other (51). Methods and results from one‐way quantum computation were also applied in statistical mechanics, where it was for example, shown that all classical spin models could be unified in a lattice gauge theory (52).

One area in which one‐way quantum computation has arguably been more successful than the circuit model is in the combination of quantum computation with quantum cryptography. Hinting at this, graph states were shown to provide the ideal structure for the development of quantum‐secret‐sharing schemes (53). The idea was taken to its fullest by the development of blind quantum computation (54). A blind computation is computation in a setting where a user wishes to perform computation on a third‐party hardware (this is closely analogous to cloud computing) in a secure way. The third party should learn nothing about the computation performed, any input or data involved in the computation, and nothing about the outcome of the computation. Blind quantum computation was first achieved in the one‐way model of quantum computation and relies upon a number of its intrinsic features.

Ideas from one‐way quantum computing have made important contributions for quantum communications, with advantages in the design of quantum repeaters (55), entanglement purification (56), and a measurement‐based approach to quantum communication itself (57).

One‐way quantum computation has also proven a valuable new lens to study some of the key foundational experiments that distinguish the quantum from the classical world. In particular, the quantum nonlocality and quantum contextuality present in the violation of Bell inequalities and the so‐called GHZ paradox could be unified as instances of one‐way quantum computation (58), and that contextuality and computation in the one‐way model are fundamentally related (59).

Cluster‐state one‐way quantum computation and its variants have been the focus of extensive experimental interest and have now been demonstrated in a variety of physical systems. Photonic quantum computing has continued to demonstrate many aspects of one‐way quantum computing, including graph‐state generation (60), active feed‐forward between measurement outcome and measurement basis selection (61), error correction (62), blind quantum computation (63), and the demonstration of quantum algorithms in the one‐way model (64). A novel approach for generating photonic cluster states – the photon “machine gun” (65) – was also recently demonstrated (66), showing that sources of entangled cluster‐state photons can be constructed. Continuous‐variable cluster states are well suited to optical generation at scale (67), and continuous‐variable cluster states have been generated over more than 10,000 modes in a single experiment (68). A major obstacle for optical lattice implementations, addressing individual atomic qubits in the lattice, has been resolved (69). Finally, this approach of quantum computation has also been implemented in ion traps (70), where the key elements of one‐way quantum computation and measurement‐based quantum error correction were demonstrated. Further details of some of the topics highlighted in this section can also be found in (71).

21.6 Outlook

In this chapter, we gave an introduction to the key ideas of one‐way quantum computation and some of the most useful mathematical techniques for describing and understanding it. The one‐way approach has provided a new paradigm for quantum computation, which casts many questions of quantum computation theory in a new light. It is leading to experimental implementations that are radically different from early ideas about how a quantum computer would operate. In addition, it is likely that there will be further physical systems in which the one‐way model offers the most achievable path to quantum computation. Not least, the success of the one‐way approach illustrates the power of novel representations of quantum information processing and should encourage us to look for other new and distinct models of quantum computation.

Acknowledgments

We would like to thank Robert Raussendorf for many insightful discussions over a number of years, which have helped to shape our perspective of one‐way quantum computation.

Hans would like to thank the Kavli Institute for Theoretical Physics (KITP) for their hospitality and support while part of this work was completed. This revised chapter is dedicated to the memory of Sean Barrett. Sean's impact on quantum computing research was profound and his loss is keenly felt.

Exercises

  1. 1 There are only two topologically distinct three‐qubit graph states. In one, the qubits form a linear three‐qubit cluster state, in the other, the qubits are connected in a triangle. Write down the stabilizer generators for these two states and hence also the full stabilizer group for each. Now show that one can transform between these two states by a local Clifford operator.
  2. 2 Prove that, to generate the stabilizer group for a images ‐qubit stabilizer subspace in an images ‐qubit system, images generators are required.
  3. 3 Consider the one‐way pattern illustrated in Figure 21.3a with angle images set to zero. Show that after the entangling CZ operation, but before the measurement, the logical operators images and images have physical representations images and images , respectively. Find the stabilizer and hence the full coset of each logical observable. When observable images is measured on the first qubit, how are the stabilizer and logical observables transformed? Hence, verify that this pattern implements a Hadamard gate.
  4. 4 Show that one‐way patterns where all input and output qubits coincide can only implement diagonal unitaries. What can one say about patterns where only some of the input and output qubits coincide?
  5. 5 Using the decomposition of an arbitrary images ‐qubit diagonal unitary images in Eq. 21.8 and by generalizing the methods in Section 21.3.8 describe a one‐way pattern that implements images requiring a total of images qubits.
  6. 6 Verify the effect of applying the “fusion” operator images to two qubits, each of which belongs to separate graph states. What happens when a projection onto the even‐parity subspace images is applied instead?
  7. 7 Consider a qubit that is prepared in an unknown state and a one‐dimensional cluster state. What is the effect of applying a fusion operator on the unknown qubit and the qubit at one end of the cluster state? How can the fusion operator be used to “input” externally provided states into a one‐way quantum computation?

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