One-way quantum computer explained
The one-way or measurement-based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.
The outcome of each individual measurement is random, but they are related in such a way that the computation always succeeds. In general, the choices of basis for later measurements need to depend on the results of earlier measurements, and hence the measurements cannot all be performed at the same time.
The hardware implementation of MBQC mainly relies on photonic devices,[1] due to the difficulty of entangling photons without measurements, and the relative simplicity of creating and measuring them. However, MBQC is also possible with matter-based qubits.[2] The process of entanglement and measurement can be described with the help of graph tools and group theory, in particular by the elements from the stabilizer group.
Definition
The purpose of quantum computing focuses on building an information theory with the features of quantum mechanics: instead of encoding a binary unit of information (bit), which can be switched to 1 or 0, a quantum binary unit of information (qubit) can simultaneously turn to be 0 and 1 at the same time, thanks to the phenomenon called superposition.[3] [4] [5] Another key feature for quantum computing relies on the entanglement between the qubits.[6] [7] [8]
In the quantum logic gate model, a set of qubits, called register, is prepared at the beginning of the computation, then a set of logic operations over the qubits, carried by unitary operators, is implemented.[9] [10] A quantum circuit is formed by a register of qubits on which unitary transformations are applied over the qubits. In the measurement-based quantum computation, instead of implementing a logic operation via unitary transformations, the same operation is executed by entangling a number
of input qubits with a cluster of
ancillary qubits, forming an overall source state of
qubits, and then measuring a number
of them.
[11] [12] The remaining
output qubits will be affected by the measurements because of the entanglement with the measured qubits. The one-way computer has been proved to be a universal quantum computer, which means it can reproduce any unitary operation over an arbitrary number of qubits.
[13] [14] [15] General procedure
The standard process of measurement-based quantum computing consists of three steps:[16] [17] entangle the qubits, measure the ancillae (auxiliary qubits) and correct the outputs. In the first step, the qubits are entangled in order to prepare the source state. In the second step, the ancillae are measured, affecting the state of the output qubits. However, the measurement outputs are non-deterministic result, due to undetermined nature of quantum mechanics: in order to carry on the computation in a deterministic way, some correction operators, called byproducts, are introduced.
Preparing the source state
At the beginning of the computation, the qubits can be distinguished into two categories: the input and the ancillary qubits. The inputs represent the qubits set in a generic
|\psi\rangle=\alpha|0\rangle+\beta|1\rangle
state, on which some unitary transformations are to be acted. In order to prepare the source state, all the ancillary qubits must be prepared in the
state:
[18] |+\rangle=\tfrac{|0\rangle+|1\rangle}{\sqrt{2}},
where
and
are the quantum encoding for the classical
and
bits:
|0\rangle=\begin{pmatrix}1\ 0\end{pmatrix}; |1\rangle=\begin{pmatrix}0\ 1\end{pmatrix}
.A register with
qubits will be therefore set as
. Thereafter, the entanglement between two qubits can be performed by applying a
gate operation.
[19] The matrix representation of such two-qubits operator is given by
CZ=\begin{bmatrix}1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 0&0&0&-1\end{bmatrix}.
The action of a
gate over two qubits can be described by the following system:
\begin{cases}
CZ|0+\rangle=|0+\rangle\\
CZ|0-\rangle=|0-\rangle\\
CZ|1+\rangle=|1-\rangle\\
CZ|1-\rangle=|1+\rangle\end{cases}
When applying a
gate over two ancillae in the
state, the overall state
CZ|++\rangle=
| |0+\rangle+|1-\rangle |
\sqrt{2 |
}turns to be an entangled pair of qubits. When entangling two ancillae, no importance is given about which is the control qubit and which one the target, as far as the outcome turns to be the same. Similarly, as the
gates are represented in a diagonal form, they all commute each other, and no importance is given about which qubits to entangle first. Photons are the most common source to prepare entangled physical qubits.
[20] [21] [22] Measuring the qubits
The process of measurement over a single-particle state can be described by projecting the state on the eigenvector of an observable. Consider an observable
with two possible eigenvectors, say
and
, and suppose to deal with a multi-particle quantum system
. Measuring the
-th qubit by the
observable means to project the
state over the eigenvectors of
:
|\Psi'\rangle=|oi\rangle\langleoi|\Psi\rangle
.The actual state of the
-th qubit is now
, which can turn to be
or
, depending on the outcome from the measurement (which is probabilistic in quantum mechanics). The measurement projection can be performed over the eigenstates of the
M(\theta)=\cos(\theta)X+\sin(\theta)Y
observable:
M(\theta)=\cos(\theta)\begin{bmatrix}0&1\ 1&0\end{bmatrix}+\sin(\theta)\begin{bmatrix}0&-i\ i&0\end{bmatrix}=\begin{bmatrix}0&e-i\ ei&0\end{bmatrix}
,where
and
belong to the
Pauli matrices. The eigenvectors of
are
|\theta\pm\rangle=|0\rangle\pmei|1\rangle
. Measuring a qubit on the
-
plane, i.e. by the
observable, means to project it over
or
. In the one-way quantum computing, once a qubit has been measured, there is no way to recycle it in the flow of computation. Therefore, instead of using the
notation, it is common to find
to indicate a projective measurement over the
-th qubit.
Correcting the output
After all the measurements have been performed, the system has been reduced to a smaller number of qubits, which form the output state of the system. Due to the probabilistic outcome of measurements, the system is not set in a deterministic way: after a measurement on the
-
plane, the output may change whether the outcome had been
or
. In order to perform a deterministic computation, some corrections must be introduced. The correction operators, or byproduct operators, are applied to the output qubits after all the measurements have been performed.
[23] The byproduct operators which can be implemented are
and
.
[24] Depending on the outcome of the measurement, a byproduct operator can be applied or not to the output state: a
correction over the
-th qubit, depending on the outcome of the measurement performed over the
-th qubit via the
observable, can be described as
, where
is set to be
if the outcome of measurement was
, otherwise is
if it was
. In the first case, no correction will occur, in the latter one a
operator will be implemented on the
-th qubit. Eventually, even though the outcome of a measurement is not deterministic in quantum mechanics, the results from measurements can be used in order to perform corrections, and carry on a deterministic computation.
CME pattern
The operations of entanglement, measurement and correction can be performed in order to implement unitary gates. Such operations can be performed time by time for any logic gate in the circuit, or rather in a pattern which allocates all the entanglement operations at the beginning, the measurements in the middle and the corrections at the end of the circuit. Such pattern of computation is referred to as CME standard pattern. In the CME formalism, the operation of entanglement between the
and
qubits is referred to as
. The measurement on the
qubit, in the
-
plane, with respect to a
angle, is defined as
. At last, the
byproduct over a
qubit, with respect to the measurement over a
qubit, is described as
, where
is set to
if the outcome is the
state,
when the outcome is
. The same notation holds for the
byproducts.
When performing a computation following the CME pattern, it may happen that two measurements
and
on the
-
plane depend one on the outcome from the other. For example, the sign in front of the angle of measurement on the
-th qubit can be flipped with respect to the measurement over the
-th qubit: in such case, the notation will be written as
, and therefore the two operations of measurement do commute each other no more. If
is set to
, no flip on the
sign will occur, otherwise (when
) the
angle will be flipped to
. The notation
can therefore be rewritten as
.
An example: Euler rotations
As an illustrative example, consider the Euler rotation in the
basis: such operation, in the gate model of quantum computation, is described as
[25] eiRX(\phi)RZ(\theta)RX(λ)
,where
are the angles for the rotation, while
defines a global phase which is irrelevant for the computation. To perform such operation in the one-way computing frame, it is possible to implement the following
CME pattern:
[26]
,where the input state
|\psi\rangle=\alpha|0\rangle+\beta|1\rangle
is the qubit
, all the other qubits are auxiliary ancillae and therefore have to be prepared in the
state. In the first step, the input state
must be entangled with the second qubits; in turn, the second qubit must be entangled with the third one and so on. The entangling operations
between the qubits can be performed by the
gates.
In the second place, the first and the second qubits must be measured by the
observable, which means they must be projected onto the eigenstates
of such observable. When the
is zero, the
states reduce to
ones, i.e. the eigenvectors for the
Pauli operator. The first measurement
is performed on the qubit
with a
angle, which means it has to be projected onto the
states. The second measurement
is performed with respect to the
angle, i.e. the second qubit has to be projected on the
state. However, if the outcome from the previous measurement has been
, the sign of the
angle has to be flipped, and the second qubit will be projected to the
state; if the outcome from the first measurement has been
, no flip needs to be performed. The same operations have to be repeated for the third
and the fourth
measurements, according to the respective angles and sign flips. The sign over the
angle is set to be
. Eventually the fifth qubit (the only one not to be measured) figures out to be the output state.
At last, the corrections
over the output state have to be performed via the byproduct operators. For instance, if the measurements over the second and the fourth qubits turned to be
and
, no correction will be conducted by the
operator, as
. The same result holds for a
outcome, as
and thus the squared Pauli operator
returns the identity.
As seen in such example, in the measurement-based computation model, the physical input qubit (the first one) and output qubit (the third one) may differ each other.
Equivalence between quantum circuit model and MBQC
The one-way quantum computer allows the implementation of a circuit of unitary transformations through the operations of entanglement and measurement. At the same time, any quantum circuit can be in turn converted into a CME pattern: a technique to translate quantum circuits into a MBQC pattern of measurements has been formulated by V. Danos et al.[27]
Such conversion can be carried on by using a universal set of logic gates composed by the
and the
operators: therefore, any circuit can be decomposed into a set of
and the
gates. The
single-qubit operator is defined as follows:
J(\theta)=
\begin{pmatrix}1&ei\ 1&-ei\theta\end{pmatrix}
.The
can be converted into a
CME pattern as follows, with qubit 1 being the input and qubit 2 being the output:
which means, to implement a
operator, the input qubits
must be entangled with an ancilla qubit
, therefore the input must be measured on the
-
plane, thereafter the output qubit is corrected by the
byproduct. Once every
gate has been decomposed into the
CME pattern, the operations in the overall computation will consist of
entanglements,
measurements and
corrections. In order to lead the whole flow of computation to a
CME pattern, some rules are provided.
Standardization
In order to move all the
entanglements at the beginning of the process, some rules of
commutation must be pointed out:
.The entanglement operator
commutes with the
Pauli operators and with any other operator
acting on a qubit
, but not with the
Pauli operators acting on the
-th or
-th qubits.
Pauli simplification
The measurement operations
commute with the corrections in the following manner:
,
where
. Such operation means that, when shifting the
corrections at the end of the pattern, some dependencies between the measurements may occur. The
operator is called signal shifting, whose action will be explained in the next paragraph. For particular
angles, some simplifications, called Pauli simplifications, can be introduced:
.
Signal shifting
The action of the signal shifting operator
can be explained through its rules of commutation:
.
The
operation has to be explained: suppose to have a sequence of signals
, consisting of
, the operation
means to substitute
with
in the sequence
, which becomes
. If no
appears in the
sequence, no substitution will occur. To perform a correct
CME pattern, every signal shifting operator
must be translated at the end of the pattern.
Stabilizer formalism
When preparing the source state of entangled qubits, a graph representation can be given by the stabilizer group. The stabilizer group
is an
abelian subgroup from the
Pauli group
, which one can be described by its generators
\{\pm1,\pmi\} x \{I,X,Y,Z\} ⊗
.
[28] [29] A stabilizer state is a
-qubit state
which is a unique eigenstate for the generators
of the
stabilizer group:
Si|\Psi\rangle=|\Psi\rangle.
Of course,
.
It is therefore possible to define a
qubit graph state
as a quantum state associated with a graph, i.e. a set
whose
vertices
correspond to the qubits, while the edges
represent the entanglements between the qubits themselves. The vertices can be labelled by a
index, while the edges, linking the
-th vertex to the
-th one, by two-indices labels, such as
.
[30] In the stabilizer formalism, such graph structure can be encoded by the
generators of
, defined as
[31] [32]
,where
stands for all the
qubits neighboring with the
-th one, i.e. the
vertices linked by a
edge with the
vertex. Each
generator commute with all the others. A graph composed by
vertices can be described by
generators from the stabilizer group:
\langleK1,K2,...,Kn\rangle
.
While the number of
is fixed for each
generator, the number of
may differ, with respect to the connections implemented by the edges in the graph.
The Clifford group
See also: Clifford gates.
The Clifford group
is composed by elements which leave invariant the elements from the Pauli's group
:
[33] l{C}n=\{U\inSU(2n) | USU\dagger\inl{P}n,S\inl{P}n\}
.The Clifford group requires three generators, which can be chosen as the Hadamard gate
and the phase rotation
for the single-qubit gates, and another two-qubits gate from the
(controlled NOT gate) or the
(controlled phase gate):
H=
\begin{bmatrix}1&1\ 1&-1\end{bmatrix}, S=\begin{bmatrix}1&0\ 0&i\end{bmatrix}, CNOT=\begin{bmatrix}1&0&0&0\ 0&1&0&0\ 0&0&0&1\ 0&0&1&0\end{bmatrix}
.
Consider a state
which is stabilized by a set of stabilizers
. Acting via an element
from the Clifford group on such state, the following equalities hold:
[34] U|G\rangle=USi|G\rangle=USiU\daggerU|G\rangle=S'iU|G\rangle
.
Therefore, the
operations map the
state to
and its
stabilizers to
. Such operation may give rise to different representations for the
generators of the stabilizer group.
The Gottesman–Knill theorem states that, given a set of logic gates from the Clifford group, followed by
measurements, such computation can be efficiently simulated on a classical computer in the strong sense, i.e. a computation which elaborates in a polynomial-time the probability
for a given output
from the circuit.
[35] [36] [37] Hardware and applications
Topological cluster state quantum computer
Measurement-based computation on a periodic 3D lattice cluster state can be used to implement topological quantum error correction.[38] Topological cluster state computation is closely related to Kitaev's toric code, as the 3D topological cluster state can be constructed and measured over time by a repeated sequence of gates on a 2D array.[39]
Implementations
One-way quantum computation has been demonstrated by running the 2 qubit Grover's algorithm on a 2x2 cluster state of photons.[40] [41] A linear optics quantum computer based on one-way computation has been proposed.[42]
Cluster states have also been created in optical lattices,[43] but were not used for computation as the atom qubits were too close together to measure individually.
AKLT state as a resource
)
AKLT state on a 2D
honeycomb lattice can be used as a resource for MBQC.
[44] [45] More recently it has been shown that a spin-mixture AKLT state can be used as a resource.
[46] See also
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