One-Way Quantum Computing Andrew Lopez
A commonly used model in the field of quantum computing is the Quantum Circuit Model. The Circuit Model can be thought of as a quantum
version of classical computing, where a computer is characterized by an quantum input state, quantum logic gates which perform operations on the input
state, and an output state. Over time other models have been developed and
are being researched, such as the Topological Quantum Model, and Adiabatic
Quantum Model. The quantum computing model which will be discussed in
this paper is called one-way quantum computation. First we will discuss how
quantum computations are performed with this one-way model, then we will
discuss the advantages the one-way model has over the circuit model, and
finally we will finish with the future of one-way quantum computing.
One-way quantum computation requires that a certain type of entangled
state, called a cluster or graph state, be set up beforehand [1]. Once the
cluster or graph state is prepared we can perform a universal quantum computation by performing a series of single-qubit measurements. Due to the
fact that the computation is performed by single-qubit measurements, this
model is inherently irreversible (one-way). The inherent irreversiblility is in
contrast with the reversibility of every gate used in the Circuit Model. Now
to take a closer look at the process, let us discuss cluster and graph states.
A graph state |Gi is associated with a graph, G, in which every qubit is
represented by a vertex and each edge represents the interaction between respective qubits. Since graphs can vary greatly in layout it is sometimes easier
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to generate an entangled state with a corresponding graph in the shape of a
square lattice. This type of state is called a cluster state. The type of entangled state you should use depends on your physics system. While cluster
states can be generated efficiently they also require many more qubits than
a graph state to perform the same computation.
Now we will briefly consider a simple example of one-way computation
that involves two qubits [2]. Before we begin let me define a few operations. The operation Uk (γ) (where k is x,y, or z) is exp(i γ2 σk ) , the “CZ”
operation is |0ih0|
N
N
Iˆ + |1ih1| σz , and the operation (θ, φ) is defined as
Uz (φ + π2 )Ux (θ)σz Ux (−θ)Uz (−φ − π2 ). Qubit 1 is prepared in an unknown
state: |ψi = α|0i + β|1i, where |0i, |1i are eigenvectors of σz . Qubit 2 is
√1 (|0i
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prepared in the known state: |+i =
+ |1i), where |+i is an eigen-
vector of σx . Now we apply a “CZ” operation to each qubit: and find that
the qubits are in an entangled state: |q1 , q2 i =
√1 (α|0i|+i
2
we perform a measurement on qubit 1 in the basis
+ β|1i|−i). Next
√1 (|0i
2
± exp(i ∗ φ)|1i)
thereby destroying qubit 1; this measurement corresponds to the measurement ( π2 , φ). From this measurement there are only two possible outcomes
(an eigenvalue of ±1). If we define a binary digit m ∈ 0, 1 to represent the
measurement outcome (−1)m , then the final state of qubit 2 can be written,
up to a phase, as σxm HUz (φ)|ψi. So we can see that the unknown state on
qubit 1 has been passed to qubit 2 without any loss of coherence. However,
if the measurement on qubit 1 returns the -1 eigenvalue, then qubit 2 will
undergo an additional Pauli transformation. This example shows that any
desired unitary transformation can be implemented only up to random but
known Pauli transformations. These unwanted Pauli transformations are
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called “by-product operators”, as they are a by-product of the randomness
of the measurements.
Since we are confident that one-way computation is viable for one singlequbit operation let’s attempt three consecutive single-qubit operations. If
we perform the two-qubit measurement protocol as described above three
times for three arbitrary angles φ1 , φ2 , φ3 , then the net unitary transformation
applied to qubit 2 will be U = Hσzm3 Uz (φ3 )Hσzm2 Uz (φ2 )Hσzm1 Uz (φ1 ), where
m1 , m2 , m3 are the binary digits for measurement 1,2, and 3 respectively. By
using a few identities we see that
U = σxm3 σzm2 σxm1 HUz ((−1)m2 φ3 )Ux ((−1)m2 φ2 )Uz (φ1 ). As we can see, the sign
of two rotations depend on two of the measurements which are random. This
would cause one to question whether this computation is deterministic. It
is not, however by appropriate choice of angles φ1 = α, φ2 = (−1)m1 β, and
φ3 = (−1)m2 γ we can obtain a deteministic unitary transformation. The
dependency that some of our angles have on previous measurements occurs
for all but one class of operations, called the Clifford group. This dependency
introduces a time-ordering to the measurements, in other words, there is a
limit on how quickly a computation can be implemented. Also since the byproduct Pauli operators are known, they can be corrected through classical
post-processing.
Stabilizer formalism is a tool use to characterize graph states and subspaces over mulitple qubits in a compact way. In the stabilizer formalism
these states and sub-spaces are described in terms of operators under which
they are invariant. A similar technique is used in standard quantum mechanics, for example, describing the atomic states by the quantum numbers.
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The formalism focuses on operators that satisfy two properties: 1) the operators stabilizes the sub-space, and 2) the operators are Hermitian members
of the Pauli group. The defnition of property 1 is as follows: An operator K
stablizes a substapce S when, for all states |ψi ∈ S, K|ψi = |ψi. The key
principle of the formalism is to identify a set of stabilizing operators that
uniquely define the state, or sub-space of interest. The stabilizer formalism
really shines when used in conjunction with the Clifford group. Application of the formalism leads to the conclusion that any stabilizer state can be
transformed to a graph state by local Clifford operations. These graph states
are, in general, not unique. Therefore given any one-way pattern containing Pauli measurements, Clifford operations can be used to find a one-way
pattern which performs the operation with fewer qubits [2].
Many recent developments have been made in the field of one-way quantum computation. In 2011, Ryuji Ukai and his collaborators demonstrated
the use of a Controlled Phase (CZ) gate for continuous-variable one-way
quantum computation on a four-mode linear cluster state [3]. The fourmode linear cluster state can be interpreted as two Einstein-Podolsky-Rosen
(EPR) paires with a CZ interaction between them. Since the CZ gate is the
gate that entangles the EPR pairs it is vital to prove that it’s implementation
is nonclassical. To that end a sufficient condition for entanglement of a twomode state is given as h∆2 (g p̂µ − x̂v )i + h∆2 (g p̂v − x̂µ )i < g for some g ∈ <.
For the experimental setup used by Ukai, it was determined that g = 3/4
gives the minimal resource requirement. As shown in Fig. 1 the condition
for entanglement was met. Ukai concludes by saying that the quality of the
CZ gate is only limited by the squeezing level of the resource state, and
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Figure 1: Fig. 1: Lines (viii) show the sufficient condition for entanglement.
Traces (vii) show the measurement results.
mentions that once a single-mode non-Gaussian gate can be experimentally
implemented then full universality of arbitrary multi-mode quantum optical
states will be realizable.
In 2013, B. A. Bell and his collaborators characterized the quantum H
gate, T gate, and CNOT gate for one-way quantum computation [4]. These
three gates form a universal set of gates and therefore allow Bell to simulate
and predict the performance of any one-way quantum quantum computation. Bell used two photonic crystal fibre sources to produce pairs of Bell
states, and then fused the pairs with a polarising beamsplitter to generate
an entangled four-photon ’star’ cluster state. At first glance, the star cluster
state seems to put us at a disadvantage. Since the cluster state is made with
photons of two different wavelengths the standard methods for generating
a larger resource state for universal quantum computation cannot be used.
However, by performing what is known as a fusion operation, star cluster
states can be combined until a resource state of desired size is generated.
This resource state will be in a hexagonal lattice which can be converted to a
square 2D lattice by using local operations, thereby confirming the resource
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state as universal. The converion of the hexagonal resource state to a square
2D lattice is not always necessary as, for one-way quantum computation, the
hexagonal lattice has an intrinsic robustness to noise.
The state was characterized and compared to the ideal cluster state
|ψstar i by carrying out state tomography. State tomography is the process
of analysing the state in all combinations of three measurement bases for
each photon: |Hi, |V i, |+i, |−i, and |Ri, |Li where |Ri, |Li are the right and
left circular polarizations. After performing all of the measurements, Bell
and his group were able to reconstruct the density matrix, ρexp . They also
calculated the fidelity of the state, F = hψstar |ρexp |ψstar i = 0.66 ± 0.01 which
is well above the threshold value of 0.5, proving that the state has genuine
four-party entanglement. Keeping in mind that the fault-tolerant threshold
is 10−2 − 10−5 , Bell calculated the error-probability per gate, for the H
gate ( > 0.33 ± 0.03), the T gate ( > 0.24 ± 0.04), and the CNOT gate
( > 0.36 ± 0.01). We can see there is much room for improvement. These
results were then used to simulate a SWAP gate by linking three CNOT gates
together. The error-probability per gate was calculated to be > 0.70 ± 0.01.
It is obvious that by simulating an operation with only three concatenated
gates, the quality of the operation is greatly reduced. The limitations of the
experiment are largely due to the multiphoton count rates and interference
visibilities of the photonic crystal fibre sources. Once the sources improve,
specifically the collection efficiency of the photons and their indistinguishability, then it will become possible to investigate experimentally larger cluster
states and more complicated one-way protocols and algorithms.
One-way quantum computation unique way to look at computation. While
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the circuit model draws inspiration from classical computation, the one-way
model cleverly uses its quantum nature to solve old problems in new ways.
The one-way model has many advantages over the circuit model such as
the one-way structure being more flexible than the circuit model due to the
necessary adaptive measurements. At the same time, however, it has just
as many disadvantages, for example the adaptive measurements introducing
a time-ordering to the system, limiting how quickly a computation can be
performed. Recent experiments show that at our current level of technology
building a one-way quantum computer to efficiently perform computations is
far off, but in time we may see one-way quantum computers used in scientific
institutions worldwide.
References:
[1] Quantum Computation and Quantum Information, by Michael A.
Nielsen and Isaac L. Chuang, Cambridge University Press.
[2] One-way Quantum Computation. Dan Browne and Hans Briegel. arXiv:quantph/060322v2 3 Oct 2006
[3] Demonstration of a Controlled-Phase Gate for Continuous-Variable OneWay Quantum Computation. Ryuji Ukai, Shota Yokoyama, et al. arXiv:1107.0514v1
[quant-ph] 4 Jul 2011
[4] Experimental characterization of universal one-way quantum computing.
B.A. Bell, M.S. Tame, et al. arXiv:1305.0212v1 [quant-ph] 1 May 2013
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