Pis’ma v ZhETF, vol. 111, iss. 9, pp. 623 - 624
© 2020
May 10
Quantum R-matrices as universal qubit gates
N. Kolganov+∗1), An. Morozov+∗×1)
+Moscow Institute of Physics and Technology, 141701 Dolgoprudny, Russia
∗Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia
×Institute for Information Transmission Problems, 127994 Moscow, Russia
Submitted 7 April 2020
Resubmitted 10 April 2020
Accepted 10
April 2020
DOI: 10.31857/S1234567820090086
Quantum computers is quite a hot topic nowadays.
versal quantum gates. In [3] it was suggested that quan-
They are a very promising device and method of solv-
tum R-matrices can indeed be used as universal quan-
ing lots of problems. The main problem of the quantum
tum gates.
computers from the practical point of view is the high
In the present papers we continue these studies with
probability of errors. Since it has quantum nature the
the goal of studying the properties of the R-matrices
states are not stable and can dissolve or change. This
as quantum gates and how different other gates can
limits the time the quantum computer can work and
be constructed from them using Solovay-Kitaev algo-
the size of the programs it can run. To deal with these
rithm [4, 5]. We construct an approximation for one-
problems the quantum correction algorithms are usually
qubit Hadamard and π/8 gates from fundamental R
used. These algorithms imply that instead of physical
and Racah matrices. The generalization to larger ma-
qubits the logical qubits, consisting of several physical
trices and higher representations is a work in progress.
ones, are considered. The physical qubits inside of the
We study the simplest topological theory which is
logical qubit are regularly entangled with each other
the Chern-Simons theory. It‘s a 3d topological gauge
thus providing an error corrections. However, the qun-
theory with Wilson-loop averages as their most inter-
tum computer with such error corrections require many
esting observables. These Wilson-loop averages for the
more physical qubits which is also a big problem at the
SU(N) gauge group are equal to HOMFLY-PT polyno-
current stage.
mials. HOMFLY-PT polynomials depend on two vari-
Another approach is to try to use qubit models where
ables A and q, which are connected to the parameters
the states are more stable. One of the approaches is to
of the theory:
make the states topological, as those are usually much
2πi
q = exp
,
A=qN.
(1)
harder to change. This leads to the idea of the topologi-
k+N
cal quantum computer. Many of the models of this quan-
The HOMFLY-PT polynomials can be calculated
tum computer behave under the laws of the topologi-
as a product of quantum R-matrices - the solutions
cal 3d Chern-Simons theory. There are different models
of Yang-Baxter equation, this is called Reshetikhin-
where the Chern-Simons is an effective theory which in
Turaev (RT) approach. One of the crucial properties
future could provide us with the topological quantum
of the R-matrix is that it acts in the same way on all el-
computer [1].
ements of the representation of the corresponding quan-
The main observables, studied in the Chern-Simons
tum group. Thus one can move to the R-matrix in the
theory are Wilson-loop averages, and this loops are
space of intertwining operators, which describes how it
usually thought to be related to quantum programs
acts on all irreducible representations in the tensor prod-
or algorithms. As we know these Wilson-loop aver-
uct of a pair of representations. The eigenvalues of such
ages are equal to the knot invariants. According to the
R-matrices are described in a very simple way for any
Reshetikhin-Turaev approach [2], these knot invariants
representation Q from the product of two representa-
can be constructed from the R-matrices. In this sense
tions R [2]:
these matrices provide an elementary building blocks
λQ = ǫQqκQ-4κRA-|R|.
(2)
from which the whole knot is constructed. In quantum
The physically meaningful Chern-Simons theories
information theory such building blocks are called uni-
has integer k and of course integer N. This means that
both q and A are roots of unity. Thus the R-matrix in
1)e-mail: nikita.kolganov@phystech.edu; andrey.morozov@itep.ru
the space of intertwining operators is unitary.
Письма в ЖЭТФ том 111 вып. 9 - 10
2020
623
624
N. Kolganov, An. Morozov
We use the R-matrices which appear for the two-
bridge knots. Namely there appear two diagonal R-
Fig. 1. First approximation for Hadamard gate, knots 74,
matrices and two Racah matrices which give the non-
923, 74, 923, 94, accuracy ǫ = 0.0068, N = 2, k = 13
diagonal matrices. In this letter we mainly discuss the
fundamental representation. In this case these matrices
are [6]:
We suggest to use the known polynomials and corre-
(
)
(
)
sponding products of R-matrices for knots up to some
q/A
1
T =
,
T =
,
number of crossings (for example 11 or 12) as a basic
-1/qA
-A
approximation. Using this we construct, as an exam-
 √
√
A
- q
Aq -1Aq
ple, approximations for Hadamard (see Fig. 1) and π/8-
1
A
S =
√
 √ q
√
,
gates E, which are the most common universal quantum
(q+q-1 )(A-A-1)
Aq -1Aq
- Aq -q
A
gates:
√
1
(
)
(
)
(Aq-A
)(Aq -qA )
q-q-1
q
1
1
1
1
0
A-A-1
A-A-1
S =
√
.
(3)
H =
√
,
E =
(4)
(Aq-1Aq )(Aq -qA )
2
1
-1
0
ei4
-q-q-1
A-A-1
A-A-1
We use these R-matrices as universal quantum gates,
We are grateful for very useful discussions to
A. Andreev, N. Tselousov, S. Mironov, A. Sleptsov, and
which are a set of elementary operations, from which the
quantum programs and algorithms are constructed. To
A. Popolitov.
This work was supported by Russian Science Foun-
use our suggest quantum gates we apply the Solovay-
Kitaev theorem [4], which says that if there is a set of
dation grant # 18-71-10073.
universal quantum gates then any program or algorithm
Full text of the paper is published in JETP Letters
represented by unitary matrix, can be approximated in
journal. DOI: 10.1134/S0021364020090027
a logarithmic time and with logarithmic number of oper-
ators. It can be proven by providing a specific algorithm
constructing such an approximation, which we refer as
1. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and
Solovay-Kitaev algorithm [5]. The pseudocode of this
S. Das Sarma, Rev. Mod. Phys. 80,
1083
(2008);
algorithm reads
arXiv:0707.1889.
2. A. Mironov, A. Morozov, and An. Morozov, JHEP 03,
Algorithm 1 Solovay-Kitaev algorithm
034 (2012); arXiv:1112.2654.
function Solovay-Kitaev (gate U, depth n)
3. D. Melnikov, A. Mironov, S. Mironov, A. Morozov,
if n = 0 then
and An. Morozov, Nucl. Phys. B 926, 491 (2018);
return Basic-Approximation (U)
arXiv:1703.00431.
else
4. A. Y. Kitaev, Russian Math. Surveys 52, 1191 (1997).
Un-1 ← Solovay-Kitaev(U, n - 1)
V, W ← GC-Decompose(U U†n-1)
5. Ch. M. Dawson and M. A. Nielsen, Quantum Info.
Comput. 6(1), 81 (2006); quant-ph/0505030.
Vn-1 ← Solovay-Kitaev(V, n - 1)
Wn-1 ← Solovay-Kitaev(W, n - 1)
6. A. Mironov, A. Morozov, and A. Sleptsov, JHEP 07,
return Vn-1Wn-1V†n-1W†
Un-1
069 (2015); arXiv:1412.8432.
n-1
Письма в ЖЭТФ том 111 вып. 9 - 10
2020