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A SWITCHING APPROACH FOR PERFECT STATE TRANSFER OVER A SCALABLE NETWORK ARCHITECTURE WITH SUPERCONDUCTING QUBITS

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An Important Problem It is often required in quantum information processing (QIP), to transfer a quantum state from one site to another. It is known that the problem of quantum state transfer with 100% fidelity, called perfect state transfer (PST) can be studied as a problem in graph theory. Therefore an architecture of quantum computation (QC) can be designed purely in terms of graph networks. It has been found that all hypercubes (Q n) as well as the Cartesian product of path graph P 3 allow PST between their antipodal vertices in time t 0 = π/2 and π/ √ 2 respectively, independent of the order n. However, one issue is that PST is not possible from any vertex to any other vertex. Another search could be from a graph that allows any number of qubits and supports PST. Here, we address both of these problems using a technique of switching of edges. We also propose a physical realization model for our architecture with superconducting qubits with tunable couplings.
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ASWITCHING APPROACH FOR PERFECT STATE TRANSFER OVER A SCALABLE NETWORK ARCHITECTURE WITH SUPERCONDUCTING QUBITS
Siddhant Singh
Indian Institute of Technology Kharagpur
ASWITCHING APPROACH FOR PERFECT STATE TRANSFER OVER A SCALABLE NETWORK ARCHITECTURE WITH SUPERCONDUCTING QUBITS
Siddhant Singh
Indian Institute of Technology Kharagpur
An Important Problem
It is often required in quantum information processing (QIP), to transfer a quantum
state from one site to another. It is known that the problem of quantum state transfer
with 100% fidelity, called perfect state transfer (PST) can be studied as a problem
in graph theory. Therefore an architecture of quantum computation (QC) can be de-
signed purely in terms of graph networks. It has been found that all hypercubes (Qn)
as well as the Cartesian product of path graph P3allow PST between their antipodal
vertices in time t0=π/2and π/2respectively, independent of the order n[1, 2].
However, one issue is that PST is not possible from any vertex to any other vertex.
Another search could be from a graph that allows any number of qubits and sup-
ports PST. Here, we address both of these problems using a technique of switching
of edges. We also propose a physical realization model for our architecture with
superconducting qubits with tunable couplings.
Proposal for PST in hypercubes (all vertices)
The probability for getting the quantum state localised at the vertex vat time tis
given by |hy|ψ(t)i|2. Graph Ghas a PST from vertex xto vertex yat finite time t0if
|hy|exp(it0A(G))|xi| = 1 (1)
where A(G)is the adjacency matrix of G. We propose the switching of sub-
hypercubes (Qd) starting with a hypercube Qn(with |Vn|=k= 2n) by the con-
struction below [4].
Construction: Let x= (x1, . . . , xn), y = (y1, . . . , yn)∈ Vn,the vertex set of Qn.
Suppose d=|{i:xi6=yi, i = 1, . . . , n}|.Then the unique induced sub-hypercube
Qdof Qnwith x, y as antipodal vertices of Qdis given by the set of vertices
Vd={z= (z1, . . . , zn)∈ Vn:zi=xi,if xi=yi,and
zi∈ {0,1}otherwise, i = 1, . . . , n}.(2)
Once the list of such vertices is identified using a quantum or classical memory, a
switching technique is proposed to put in place to create an induced sub-hypercube
Qdof Qnsuch that x, y are antipodal vertices of Qd. The switching technique in-
volves tuning off the coupling strength of all the edges in Qnthat do not belong
to the induced sub-hypercube Qd.Indeed, once the vertex set Vdis determined,
deactivate all the couplings that incident to any vertex in Vn\ Vd.This can be ac-
complished in polynomial time with the aid of classical or quantum memory.
𝒬𝑛=4 ≡ 𝒬𝑑=2
𝒬𝑛=4 ≡ 𝒬𝑑=1
𝒬𝑛=4 ≡ 𝒬𝑑=3
𝒬𝑛=4
𝑣1𝑣2
𝑣3𝑣4
𝑣5𝑣6
𝑣7𝑣8
𝑣9𝑣10
𝑣11 𝑣12
𝑣13 𝑣14
𝑣15 𝑣16
Fig. 1: Visual representation of the switching process. Green represents switched-on edges and gray indicates
switched-off edges. Isolated vertices are in dark-gray. Any hypercube Qn(here n= 4), under switching can realise
various embedded sub-hypercubes (Qneffectively identical to Qdfor PST dynamics).
PST for any number of qubits (2n< k 2n+1)
We propose a network that enables PST for any number of qubits k, with 2n< k 2n+1 for
any n. PST in hypercubes is optimal, that is, one-step task in minimal time t0. For a general k,
therefore, the next possible optimal time for PST is a two-step PST at most. We present such
a graph construction which performs PST in maximum of two steps, with a total time 2t0[3].
Therefore, now the PST follows in two steps as
xzy(3)
with an intermediate node z. It can proved by partitioning the vertices as k= 2n+m, where
additional nodes need to be connected to an existing hypercube. If we follow the binary labeling
in sequence for all vertices (with the length n+ 1), then an interesting property arises. If the
desired vertices belong to this complete hypercube of 2nvertices, then a PST is possible.
However, if one vertex belongs to the complete hypercube and the other in the additional nodes,
the PST can be performed by a 2-step PST process in time 2t0. And the same argument follows
for all partitions within this set of mqubits. Therefore, a PST is possible between any qubits
for any arbitrary number of qubits in a maximum time of 2t0. And hence, the scalability of this
protocol.
The graph for this PST scheme can be understood with an example. Adapt the labelling of
Qn+1 and start with the vertex v1= (000 . . . 0) -n+ 1 times. Then do the binary addition with
(000 . . . 01) for the next vertex v2and so on. Therefore, the next vertex is v2= (000 . . . 01), and
so on until vk. Connect the vertices with edges which have a Hamming distance of unity. Then
it can be seen from results in [3] that there always exists a special vertex zsuch that PST can
be performed from xto zto y, if not directly between xand y.
Physical realization with superconducting qubits
Our task is to show support for switching of the hypercube Qnto Qn≡ Qdas desired for
any pair of chosen vertices of Qnfor the task of PST. Physical key requirements of our ar-
chitecture are the following: (a) nnearest-neighbour (NN) interactions to realize any general
Qnhypercube, (b) Since distant qubits are connected, implementation is not possible in planar
integration, Q4on wards. Three-dimensional (3D) integration is needed, (c) Tunable (switch-
able) edges as couplings for each pair of nodes (qubits), and (d) High fidelity control over the
processor.
(c)
Fig. 2: (a) Couplings involved between a pair of interacting qubits, forming an edge in the hypercube Qn. (b) Network of four qubits
forming Q2. Each ancillary coupler is associated with every edge which is controlled in the experiment. (c) Variation of the dynamic
tunable coupling ˜
Jw.r.t. the detuning for each qubit. There exists a cutoff value which is always guaranteed.
We consider a general system that consists of 2nqubits for Qnwith exchange coupling between nearest
qubits (which have an edge between them). In addition, for having a switchable coupling an extra qubit
between them is also needed [5]. The total Hamiltonian including both physical and ancillary qubits (n2n1)
is given by:
HQn
~=1
2
2n
X
i=1
ωiσz
i+1
2X
hi,ji
ωCijσz
Cij +X
hi,ji
giσx
iσx
Cij +X
hi,ji
gijσx
iσx
j(4)
We consider the dispersive regime in which the qubits are well detuned, i.e. gi |i| ∀i, with i=
ωiωCij <0the qubit-ancilla detuning. Therefore, we can use a perturbation theory in gi/i. We use the
Schrieffer-Wolff unitary transformation (SWT) USW =eη. In our case,
USW = exp X
hi,jihgi
iσ+
iσ
Cij σ
iσ+
Cij+gi
Σiσ+
iσ+
Cij σ
iσ
Ciji!(5)
After the SWT we end up with the effective qubit-qubit interaction Hamiltonian:
˜
V
~=X
hi,ji
˜
Jij σ+
iσ
j+σ
iσ+
j(6)
where the effective tunable coupling between any two qubits is given by:
˜
Jij gigj
21
i
+1
j1
Σi1
Σj+gij 1
2"ω2
∆Σηij + 1#Cij
pCiCj
ω(for indentical qubits)(7)
The hopping term is tunable by setting the desired couplings and detunings. ˜
Jij can be altered negative
when ancilla coupler frequency is reduced or changed to positive when this frequency is escalated. There-
fore, we have some ωoff
Cij such that ˜
Jij(ωoff
Cij) = 0 within the bandwidth of each coupler. Thus, we obtain the
switchable edges with ωCij as the parameter.
Fidelity bound for imperfect implementation
The bound on the effective fidelity when the detuning parameters are not exact was analytically found as
F 1
X
m=1 O(kEkm
F).(8)
where Ecaptures the effect of deviation from the ideal value. Using this bound, if maximum deviation in ˜
Jij
for each edge is ±0.5%, we have F>97.43% for the hypercube Q4.
References
[1] Matthias Christandl et al. “Perfect State Transfer in Quantum Spin Networks”. In: Phys. Rev. Lett. 92 (18 May 2004),
p. 187902.
[2] Matthias Christandl et al. “Perfect transfer of arbitrary states in quantum spin networks”. In: Phys. Rev. A 71 (3 Mar. 2005),
p. 032312.
[3] Siddhant Singh. A switching approach for perfect state transfer over a scalable and routing enabled network architecture
with superconducting qubits. 2020. arXiv: 2007.02682 [quant-ph].
[4] Siddhant Singh et al. Perfect state transfer on hypercubes and its implementation using superconducting qubits. 2020. arXiv:
2011.03586 [quant-ph].
[5] Fei Yan et al. “Tunable Coupling Scheme for Implementing High-Fidelity Two-Qubit Gates”. In: Phys. Rev. Applied 10 (5 Nov.
2018), p. 054062.
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