Work-sharing of qubits in topological error corrections
Tetsufumi Tanamoto Hayato Goto


 


 tetsufumi.tanamoto@toshiba.co.jp





Corporate R & D center, Toshiba Corporation,
Saiwai-ku, Kawasaki 212-8582, Japan





Topological error-correcting codes, such as surface codes and color codes, are promising because quantum operations are realized by two-dimensionally (2D) arrayed quantum bits (qubits).
However, physical wiring of electrodes to qubits is complicated, and 3D integration for the wiring requires further development of fabrication technologies.
Here, we propose a method to reduce the congestion of wiring to qubits by just adding a SWAP gate after each controlled-NOT (CNOT) gate.
SWAP gates exchange roles of qubits. Then, the roles of qubits are shared 
between different qubits. 
We found that our method transforms the qubit layout and reduces the number of qubits that cannot be accessed two-dimensionally.
We show that fully 2D layouts including both qubits and control electrodes can be achieved for surface and color codes of minimum sizes. 
This method will be beneficial to simplifications of fabrication process of quantum circuits
in addition to improvements of reliability of qubit system.

03.67.Lx, 03.67.Mn, 73.21.La
Work-sharing of qubits in topological error corrections
    Hayato Goto
    December 30, 2023
=======================================================








§ INTRODUCTION

Solid-state quantum computers <cit.> 
have made significant progress recently 
in experiments of quantum error corrections <cit.>. 
Topological  error-correcting codes,  such as surface codes <cit.> and color codes <cit.>, 
have been intensively investigated because quantum operations are realized
through physical interactions between nearest neighboring qubits.
However, physical wiring of electrodes to qubits is complicated and 3D integration for the wiring
requires further development of fabrication technologies <cit.>.
The complexity of wiring is mainly attributable to connections 
to syndrome-qubits from data-qubits.
Although quantum annealing machines based on superconducting qubits have already been developed <cit.>,
in order to realize quantum computation, accurate control of quantum error correction is
necessary.

Here, we propose a method to reduce the congestion of wiring to qubits.
We just add a SWAP gate <cit.> after each controlled-NOT (CNOT)  gate.
Because SWAP gates exchange roles of qubits, the roles of syndrome measurements are shared 
between different qubits. 
This transforms the qubit layout and reduces the number of qubits that cannot be accessed two-dimensionally.
In particular, we show that fully 2D layouts including both qubits and control electrodes 
can be achieved for surface and color codes of minimum sizes.
Because a CNOT gate is indispensable to every quantum computer,
the work-sharing of qubits by SWAP gates will be applicable to 
general quantum circuits to improve uneven workloads of qubits. 







Topological codes are based on the stabilizer formalism <cit.>,
where a parity check process is carried out by summing  `1' of 
data-qubits by using CNOT operations.
Errors can be detected by change of the parity. 
As an example, when a wave function of four data-qubits is 
given by 
|Ψ =∑_i_1,i_2,i_3,i_4=0,1 a_i_1i_2i_3i_4|i_4i_3i_2i_1 and 
a syndrome-qubit is initialized as |0, 
the measurement process of a Z-type check operator (Z-check) is given by |Ψ|0→∑ a_i_1i_2i_3i_4|i_4i_3i_2i_1 | i_4 ⊕ i_3 ⊕ i_2 ⊕ i_1
(Fig.1(a)), where ⊕ denotes summation modulo 2.
Similarly, the measurement process of a X-type check operator (X-check) is illustrated in Fig.1(b) (|+≡ [|0+|1]/2).
When the number of data-qubits increases, 
the corresponding number of wirings to a single syndrome-qubit increases.
Physical wires have a finite width and when many wires are arranged closely, 
the crosstalk problem also appears <cit.>. Thus, it is desirable to avoid the concentration 
of wires in a small space.

Using local interactions between nearest-neighbor qubits on a two-dimensional (2D) physical plane,
surface codes <cit.> and color codes <cit.> 
have a high tolerance against errors.
However, control gate electrodes 
are not generally placed in the same physical plane as qubits, 
and vertical access to qubits is unavoidable.
Figures 1(c) and 1(d) show Z-checks and X checks of the 7-qubit color code, respectively.
The three syndrome-qubits `A-C' are connected to seven data-qubits,
where the data-qubit `g' should be accessed from the vertical direction 
by stacking an additional wiring layer.
Figure 1(e) shows a distance-3 surface code, 
where Z-checks and X-checks are performed simultaneously <cit.>.

The code-distance d is the measure of a code in which 
⌊ (d-1)/2 ⌋ physical errors can be corrected by repeated measurements, 
and corresponds to the array size 
in the surface code.

In this code, the inner nine qubits should be  accessed vertically <cit.>.
Thus, many stabilizer codes lead to a concentration of 
wires to syndrome-qubits and the complexity of wiring between qubits is unavoidable <cit.>.
Because qubits are sensitive to decoherence, the fabrication process 
of wiring is particularly difficult in view of the state-of-the-art techniques.



Our method for reducing wiring congestion is to just add a SWAP gate after each CNOT gate in stabilizer measurements.
SWAP gates are widely used in quantum operations and exchange 
qubit states without changing system entanglement <cit.>.
The combination of a SWAP gate and a CNOT gate (hereafter a `CNOT+SWAP gate') 
shifts the role of syndrome-qubits,
and reduces the congestion of wiring. 
The replacement of CNOT gates by CNOT+SWAP gates is effective in 
stabilizer measurements because syndrome measurements are repeated many times.
We show that this method changes qubit layouts, and importantly, for codes of small sizes, 
wiring can be set on the same physical plane as qubit array. 


The insertion of a SWAP gate in each CNOT gate imposes no overhead
but rather it is advantageous
when physical interactions between qubits are XY interactions <cit.>. 
The CNOT+SWAP gate can be performed directly by the XY interaction (iSWAP) with single-qubit rotations
(Fig.2(a)) <cit.>.
On the other hand, a CNOT gate needs two iSWAP gates,
which makes the CNOT operation complicated and more fragile.
Thus, in the case of the XY interaction,
the replacement of CNOT gates by CNOT+SWAP gates makes quantum operations more reliable
and saves operation time <cit.>.
Moreover, high-precision iSWAP gates are experimentally feasible <cit.>.







This paper is organized as follows:
In Sec. II we show an example of 
the application of replacement of CNOT gate by CNOT+SWAP gate 
to fundamental stabilizer measurement.
In Sec. III, we show layout changes of a minimum 7-qubit color 
code and the second smallest color code by our method.
In Sec. IV, we first show layout changes 
of standard surface codes. Next we consider the application of our 
method to a rotated surface code, and we also show results of numerical simulations 
of logical error probabilities. 
Sec. V is devoted to a conclusion. 
In Appendix, we show  detailed explanations of the numerical simulation of Sec. IV  and 
a process of fault tolerant 7-qubit color code.




§ STABILIZER CODE

The effect of the insertion of a SWAP gate after each CNOT gate is 
easily understood when we apply this to 
the single stabilizer measurement shown in Figs. 1(a) and 1(b),
the results of which are shown in Figs. 2(b) and 2(c), respectively.
For the Z-check, we have

    |Ψ|0→∑_i_1,i_2,i_3,i_4=0,1 a_i_1i_2i_3i_4|i_4i_3i_2i_1 |i_1 
       →   ∑ a_i_1i_2i_3i_4|i_4,i_3, i_2 ⊕ i_1, i_2 |i_1 
       →   ∑ a_i_1i_2i_3i_4|i_4,i_3 ⊕ i_2 ⊕ i_1,i_3,i_2 |  i_1 
       →   ∑ a_i_1i_2i_3i_4|i_4 ⊕ i_3 ⊕ i_2 ⊕ i_1,i_4,i_3,i_2 | i_1

We can see that the SWAP gates shift the roles of qubits one by one, and finally if we regard qubit `4' 
as a new syndrome-qubit, we obtain the same wave function |Ψ as that of Fig. 1(a).
As shown in Fig. 2(b), the qubit layout is now transformed to a one-dimensional qubit array.
The same thing holds for the X-check (Fig. 2(c)):

    |Ψ|+   →   ∑ a_i_1i_2i_3i_4 |+, i_4 ⊕ +, i_3 ⊕+, i_2 ⊕ + | i_1⊕ +

If we measure qubit `4', we obtain the same wave function |Ψ as that in Fig. 1(b). 
Thus, we can avoid the congestion of wiring to one syndrome-qubit by just inserting SWAP gates. 
After the Z-checks in Fig. 2(b), when 
qubit `4'  is initialized to a |+ state, and the process of 
Fig. 2(b) is reversed from the green operation to the black operation, we can restore 
the qubits to their original roles (closed circle is back to the syndrome-qubit).
Thus, serial operations of X-checks following Z-checks can be repeated,
and the role of syndrome qubit is shared between the two end qubits.





§ COLOR CODE

Let us show the effect of the replacement of CNOT gates by CNOT+SWAP gates
in the 7-qubit color code,
which is the minimum color code (code distance 3)
 <cit.>.
New connections between qubits are determined such that each step 
reproduces the same qubit states 
as those of Figs. 1(c) and 1(d).
Figures 2(d) and 2(e) show the results of applying CNOT+SWAP gates, 
instead of CNOT gates, to the 7-qubit color code.
This replacement transforms the qubit layout to a quasi-one-dimensional layout,
where 2D access to all qubits is possible (Fig. 2(f)), and
consequently no additional 3D layer for wiring is needed.
Figure 2(e) is carried out after Fig. 2(d),
where the order of CNOT+SWAP gates 
is reversed between Figs. 2(d) and 2(e). 
That is, we can use the same connections between the Z-checks and X-checks,  
and syndrome measurements, 
in which the role of syndrome-qubits is shared in five qubits (qubits `A', `B', `C', `a' and `b'), can be repeated.
The application of our method to the next larger code (code distance 5) also 
reduces the number of qubits that cannot be accessed two-dimensionally 
from 5 to 2 qubits as shown in Fig. <ref>.

So far, the stabilizer measurements have been
assumed to be error-free. 
Otherwise, unexpected errors propagate through 
the stabilizer measurements for the color codes.
The fault-tolerant version <cit.> of the 7-qubit code is described in Appendix.

















§ SURFACE CODE

Next, we apply our idea to the standard surface code as shown in Fig. <ref>.
In each step, the quantum state of Fig. <ref> coincides with that of the standard state 
of Fig. 1(e). 
Figure <ref>(a) shows the new connections between qubits of a distance-3 surface code
after the replacement of CNOT gates by CNOT+SWAP gates.
When the connected qubits are rearranged closely, 
a new layout of qubits arises, as illustrated in Fig. <ref>(b),
where 20 qubits share the role of the syndrome measurement.
The number of the qubits
to which two-dimensional access is impossible 
is reduced to three (qubits `14', `22' and `42') 
from nine (=3×3 in Fig. 1(e)).
Similarly, the number of the qubits that 
cannot be accessed two-dimensionally for a distance-5 surface code is reduced from 49 (=7×7) to 35 (=7×5),  
as shown in Fig <ref>(c) and Fig. <ref>(d), respectively. 
In general, the number of two-dimensionally inaccessible qubits for a distance-d surface code 
is reduced from (2d-3)^2 to (2d-3)×(2d-5) by our method.


For distance-d codes, a set of syndrome measurements 
is repeated more than d times. 
The next processes of Figs. <ref>(b),(d) are
carried out by reversing both the order of the connections and the directions of arrows. 
That is, the green connections are carried out first and the black one last, 
reversing the directions of the interactions.
After the reversing process, the next connections are the same as Figs. <ref>(a),(c).
  



Rotated surface codes are known to be efficient surface codes <cit.>.
In the case of distance-3 surface codes, 
the number of qubits is reduced from 25 (Fig. 1(e)) to 17 (Fig. <ref>(a)).
The order of operations is crucial for the fault-tolerance <cit.>.
Figure <ref>(b) shows the result of the replacement of CNOT gates by CNOT+SWAP gates. 
12 qubits share the role of the syndrome measurements.
Since the central qubit cannot be accessed two-dimensionally as before, 
we introduce a technique for cutting the connections,
as a result of which a fully 2D layout is achieved. 
The cut technique is carried out by the additional five processes shown 
in Figs. <ref>(c)-(g) with ancilla qubits. 
(These operations are performed following the fourth time step shown in Fig. <ref>(b).)  
The ancilla qubits are initialized to |0 states and used to hold the quantum states 
of the qubits `0' and `14'.
By serial application of CNOT+SWAP gates, 
the same operations as those without the cut 
can be realized.
To demonstrate the usefulness of the cut technique,
we evaluated the logical error probabilities of the distance-3 rotated surface code
(Figs. <ref>(h),(i)) by numerical simulations <cit.> 
(see Appendix for details).
The simulation results shown in Figs. <ref>(h) and (i)
not only prove the fault-tolerance but also 
show substantial reduction of error probabilities.
New rotated surface code layout with electrodes is 
shown in Fig. <ref> where 
each electrode represents a set of wiring lines as used in Ref. <cit.>. 











§ CONCLUSION

Replacement of CNOT gates by CNOT+SWAP gates which corresponds to  iSWAP gates 
changes layouts of quantum error-correcting codes and relaxes the congestions of wiring 
between qubits. Importantly, we showed that fully 2D layouts including both qubits and control electrodes 
can be achieved for surface and color codes of minimum sizes.
By the work-sharing, the concentration of 
workloads to specific qubits can be relaxed, and the degradation of  the qubits will be 
mitigated.  This will improve the reliability of a quantum circuit.
Moreover, it is considered that simple 2D circuits are important for 
initial development phases of experiments.
In general, it is not easy to operate circuits as expected, 
because some mistakes are often found in designs after fabrications. 
Thus, repeated fabrications of 
chips are necessary.  Simple circuit layouts are beneficial for 
a reduction of fabrication period as well as fabrication process.



 We thank A. Nishiyama, M. Koyama, H. Hieda and S. Yasuda for discussions.




§ SIMULATION METHOD.

To show the usefulness of the cut technique, 
we performed numerical simulations and evaluated the performance. 
Figure 4(h) shows the results where single-qubit error probabilities (p_1) 
are comparable to the two-qubit error probability (p_2) such as p_1=p_2/2. 
Figure 4(i) shows the results 
where the single-qubit errors are rare compared to the two-qubit errors such as p_1=0. 
Here, we assume that the two-qubit error is an error during a 
CNOT+SWAP operation.
The curves in Figs. 4(h) and 4(i) are the fits to the simulation results with a function form of α p_2^2, 
where α is a single fitting parameter. 
These excellent fits mean that any single-qubit errors have been corrected 
and therefore prove the fault tolerance of the syndrome measurements with the cut technique. 
It is also notable that the logical error probabilities are comparable to those in Ref. <cit.>, 
where the cut technique is not used. 
Thus, we conclude that the cut technique is useful 
to realize fully accessible 2D layouts without spoiling performance.

In this simulation, we initially prepare error-free logical |0_L or |+_L 
and repeat syndrome measurements. 
In Fig. <ref>, the next process is 
carried out by reversing the order of the connections 
such that the cut is carried out first and the black one last.
At the end of each round of syndrome measurements, 
we perform recovery operations according to the measurement results,
where we estimate error positions with the lookup table decoder 
designed for the present case (see Tables 1-4). 
After the recovery, we estimate the logical error probabilities 
by error-free measurements and decoding the results. 
The error model assumed here is the standard depolarizing noise model: One of the three single-qubits 
Pauli errors occurs with probability p_1/3 on each idle qubit, 
after each initialization to |0_L and each Hadamard gate, 
and before each computational-basis measurement; 
one of the fifteen two-qubit Pauli errors occurs with probability p_2/15 
after each CNOT+SWAP gate. For this simulation, 
we used the same stabilizer simulator as Ref. <cit.>. 
From the simulation results, we estimated logical error probabilities per round with the results of 40 rounds. 


















§  FAULT-TOLERANT 7-QUBIT CODE.

One way to meet the condition of fault tolerance <cit.> 
is to use cat states in the stabilizer measurements <cit.>.
Figure <ref>(a) 
shows a direct application of the four-qubit cat-state
[|0|0 |0 |0 +|1|1|1|1]/√(2) to the color code  in Z-check. 
(Creation of the cat state is described in Fig. <ref>.)
The connections between qubits change when it is compared with Fig. 2(d) in the text,
because at most four data-qubits can access the cat state simultaneously.
Note that the connections A4-A3-B4-C1-C4-A4 constitute a closed loop, and consequently
the qubit `g' cannot be accessed in the same physical plane.

By cutting the connection 
between `A4' and `C4',
we can realize the fault-tolerant 7-qubit color code where 
all qubits can be accessed two-dimensionally.
Three ancilla qubits that 
interact with the qubits `A3', `B4', `C1' are introduced to store the quantum states of 
those qubits. 
The interaction between `A4' and `C4' with the cut technique is carried out 
by connecting qubit pairs among `A3', `B4', and `C1' one by one.
Additional 11 steps are required (see Fig. <ref>).  







































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