Supplementary material: Space-Time Duality between Quantum Chaos and
Non-Hermitian Boundary E
ff
ect
Tian-Gang Zhou,
1,
∗
Yi-Neng Zhou,
1,
∗
Pengfei Zhang,
2,
†
and Hui Zhai
1,
‡
1
Institute for Advanced Study, Tsinghua University, Beijing,100084, China
2
Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, California 91125, USA
(Dated: May 1, 2022)
In this supplementary material, we present details of the space-time duality and the choice of the non-
Hermitian and Hermitian models separately.
I. THE SPACE-TIME DUALITY OF THE TWO-QUBIT GATE
First of all, the basic element of the space-time duality can be revealed by the two-qubit gate evolution.
As demonstrated in the Fig. 1(a, b), the two-qubit gate evolving along the temporal direction is defined as:
ˆ
u
=
exp
(
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
)
exp
(
iJ
x
( ˆ
σ
x
r
+
ˆ
σ
x
r
+
1
)
)
exp
(
ih
( ˆ
σ
z
r
+
ˆ
σ
z
r
+
1
)
+
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
)
.
(1)
To apply the space-time duality, we evaluate the matrix elements of the two-qubit gate ˆ
u
in the ˆ
σ
z
eigen-basis
|
s
r
,
t
〉
. We calculate each part separately in the subsec. I A, I B and summarize the result in the subsec. I C.
A. The space-time dual of the
ˆ
σ
x
r
term
For the exp(
iJ
x
ˆ
σ
x
r
) term, the matrix element reads as:
〈
s
r
,
t
+
1
|
exp(
iJ
x
ˆ
σ
x
r
)
|
s
r
,
t
〉≡
k
exp(
i
̃
J
z
s
r
,
t
s
r
,
t
+
1
)
,
(2)
where the classical variable
s
r
,
t
can choose from either
+
1 or
−
1 and
k
2
=
i
2
sin(2
J
x
)
,
̃
J
z
=
−
π
4
+
i
2
ln(tan
J
x
)
.
(3)
Inspired by the spirit of the space-time duality, we exchange the roles of the
t
and
r
with ̃
r
≡
t
,
̃
t
≡
r
.
Then we introduce a new classical variable ̃
s
. Specifically, we have
s
r
,
t
=
s
̃
t
,
̃
r
≡
̃
s
̃
r
,
̃
t
,
(4)
2
where ̃
s
̃
r
,
̃
t
is defined as a new variable swapping the first and second subscripts of
s
̃
t
,
̃
r
. Using this definition,
the matrix elements can be equivalently expressed as
〈
s
r
,
t
+
1
|
exp(
iJ
x
ˆ
σ
x
r
)
|
s
r
,
t
〉≡
k
exp(
i
̃
J
z
s
r
,
t
s
r
,
t
+
1
)
=
k
exp(
i
̃
J
z
̃
s
̃
r
,
̃
t
̃
s
̃
r
+
1
,
̃
t
)
=
k
〈
̃
s
̃
r
,
̃
t
,
̃
s
̃
r
+
1
,
̃
t
|
exp(
i
̃
J
z
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
)
|
̃
s
̃
r
,
̃
t
,
̃
s
̃
r
+
1
,
̃
t
〉
.
(5)
B. The space-time dual of the
ˆ
σ
z
r
term
Similarly, for the exp(
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
) term:
〈
s
r
,
t
,
s
r
+
1
,
t
|
exp(
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
)
|
s
r
,
t
,
s
r
+
1
,
t
〉
=
exp(
iJ
z
s
r
,
t
s
r
+
1
,
t
)
=
exp(
iJ
z
̃
s
̃
r
,
̃
t
̃
s
̃
r
,
̃
t
+
1
)
≡
k
′
〈
̃
s
̃
r
,
̃
t
+
1
|
exp(
i
̃
J
x
ˆ
σ
x
̃
r
)
|
̃
s
̃
r
,
̃
t
〉
,
(6)
with
k
′
2
=
2
i
sin(2
̃
J
x
)
,
̃
J
x
=
arctan(
−
ie
−
2
iJ
z
)
.
(7)
C. The space-time dual of the
ˆ
u
Combining the ˆ
σ
z
r
term and the ˆ
σ
x
r
term together, we express the two-qubit gate ˆ
u
in the basis of the new
classical variable ̃
s
〈
s
r
,
t
+
1
,
s
r
+
1
,
t
+
1
|
ˆ
u
|
s
r
,
t
,
s
r
+
1
,
t
〉
=
〈
s
r
,
t
+
1
,
s
r
+
1
,
t
+
1
|
exp(
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
) exp(
iJ
x
( ˆ
σ
x
r
+
ˆ
σ
x
r
+
1
)) exp(
ih
( ˆ
σ
z
r
+
ˆ
σ
z
r
+
1
)
+
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
)
|
s
r
,
t
,
s
r
+
1
,
t
〉
=
exp(
iJ
z
s
r
,
t
+
1
s
r
+
1
,
t
+
1
)
k
exp(
i
̃
J
z
s
r
,
t
s
r
,
t
+
1
)
k
exp(
i
̃
J
z
s
r
+
1
,
t
s
r
+
1
,
t
+
1
) exp(
ih
(
s
r
,
t
+
s
r
+
1
,
t
)
+
iJ
z
s
r
,
t
s
r
+
1
,
t
)
=
exp(
iJ
z
̃
s
̃
r
+
1
,
̃
t
̃
s
̃
r
+
1
,
̃
t
+
1
)
k
exp(
i
̃
J
z
̃
s
̃
r
,
̃
t
̃
s
̃
r
+
1
,
̃
t
)
k
exp(
i
̃
J
z
̃
s
̃
r
,
̃
t
+
1
̃
s
̃
r
+
1
,
̃
t
+
1
) exp(
ih
( ̃
s
̃
r
,
̃
t
+
̃
s
̃
r
,
̃
t
+
1
)
+
iJ
z
̃
s
̃
r
,
̃
t
̃
s
̃
r
,
̃
t
+
1
)
=
k
2
exp(
i
̃
J
z
̃
s
̃
r
,
̃
t
+
1
̃
s
̃
r
+
1
,
̃
t
+
1
) exp(
iJ
z
̃
s
̃
r
+
1
,
̃
t
̃
s
̃
r
+
1
,
̃
t
+
1
) exp(
iJ
z
̃
s
̃
r
,
̃
t
̃
s
̃
r
,
̃
t
+
1
) exp(
ih
( ̃
s
̃
r
,
̃
t
+
̃
s
̃
r
,
̃
t
+
1
)) exp(
i
̃
J
z
̃
s
̃
r
,
̃
t
̃
s
̃
r
+
1
,
̃
t
)
=
k
2
k
′
2
〈
̃
s
̃
r
,
̃
t
+
1
,
̃
s
̃
r
+
1
,
̃
t
+
1
|
ˆ
v
|
̃
s
̃
r
,
̃
t
,
̃
s
̃
r
+
1
,
̃
t
〉
.
(8)
Here, the fourth line is derived by rewriting
s
as the new classical variable ̃
s
, and the fifth line rearranges
the terms in the row above. Followed by these steps, ˆ
v
is defined as
ˆ
v
=
exp(
i
̃
J
z
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
) exp(
i
̃
J
x
( ˆ
σ
x
̃
r
+
1
+
ˆ
σ
x
̃
r
)) exp(
ih
( ˆ
σ
z
̃
r
+
ˆ
σ
z
̃
r
+
1
)
+
i
̃
J
z
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
)
.
(9)
Thus, we obtain the form of ˆ
v
which is dual to the two-qubit gate ˆ
u
considered at the beginning of this
section.
3
(c)
(d)
r
t
t
~
r
~
L
r
L
t
L
r~
u
^
v
^
~
L
t
(a)
(b)
i
1
i
2
j
1
j
2
r
t
t
~
r
~
i
1
i
2
j
1
j
2
FIG. 1: Schematic of the space-time duality of quantum circuit. To be concrete, we consider more general case and the
convention is slightly di
ff
erent with the main text. (a, b): The space-time duality between two-qubit quantum circuits
ˆ
u
(left) and ˆ
v
(right) with
v
i
2
,
j
2
i
1
,
j
1
=
u
j
1
,
j
2
i
1
,
i
2
. Green and blue box respectively represent two-qubit gate
e
iJ
z
ˆ
σ
z
1
ˆ
σ
z
2
and
e
i
̃
J
z
ˆ
σ
z
1
ˆ
σ
z
2
.
Yellow and orange box respectively represent the combination of the single-qubit gates
e
iJ
x
ˆ
σ
x
e
ih
ˆ
σ
z
and
e
i
̃
J
x
ˆ
σ
x
e
ih
ˆ
σ
z
, with
the relation between
J
z
and
̃
J
z
, and relation between
J
x
and
̃
J
x
given by Eq. (3) and (7). (c, d): The space-time duality
between two general operators
ˆ
U
and
ˆ
V
, with
L
̃
r
=
L
t
and
L
̃
t
=
L
r
. Here we illustrate the
ˆ
U
and
ˆ
V
in the form of
brick-walls, in preparation for the prove in the sec. II.
II. THE SPACE-TIME DUALITY OF TRACE OF UNITARY EVOLUTION
We further derive the space-time duality of the trace of a unitary evolution. Illustrated in the Fig. 1(c,d),
we consider the one-dimensional quantum circuit with length
L
r
, evolving along the temporal direction with
L
t
steps. The time evolution of the quantum circuit reads as
ˆ
U
=
exp(
i
∑
L
r
r
∈
odd
J
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
) exp(
iJ
x
∑
L
r
r
=
1
ˆ
σ
x
r
) exp(
ih
∑
L
r
r
=
1
ˆ
σ
z
r
+
iJ
z
∑
L
r
r
∈
odd
ˆ
σ
z
r
ˆ
σ
z
r
+
1
)
t
=
odd
exp(
i
∑
L
r
r
∈
even
J
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
) exp(
iJ
x
∑
L
r
r
=
1
ˆ
σ
x
r
) exp(
ih
∑
L
r
r
=
1
ˆ
σ
z
r
+
iJ
z
∑
L
r
r
∈
even
ˆ
σ
z
r
ˆ
σ
z
r
+
1
)
t
=
even
.
(10)
Firstly, all evolutions are with periodic boundary condition(PBC). Secondly, we distinguish the even and
odd sites here, since the nearest neighbor Ising interaction exp
(
i
∑
L
r
r
=
1
(
J
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
))
is treated as two layers
of two-qubit gates with brick-wall structure, depicted in the Fig. 1(a,c). By applying the same technique in
sec. I, we insert ˆ
σ
z
eigen-basis
|
s
r
,
t
〉
in the trace of the unitary evolution with
L
t
steps, and then perform the
4
space-time rotation
Tr
L
r
(
ˆ
U
L
t
)
=
L
t
∏
t
∈
odd
exp(
iJ
z
L
r
∑
r
∈
odd
s
r
,
t
+
1
s
r
+
1
,
t
+
1
)
k
L
r
exp(
i
L
r
∑
r
=
1
̃
J
z
s
r
,
t
s
r
,
t
+
1
) exp(
ih
L
r
∑
r
=
1
s
r
,
t
) exp(
iJ
z
L
r
∑
r
∈
odd
s
r
,
t
s
r
+
1
,
t
)
L
t
∏
t
∈
even
exp(
iJ
z
L
r
∑
r
∈
even
s
r
,
t
+
1
s
r
+
1
,
t
+
1
)
k
L
r
exp(
i
L
r
∑
r
=
1
̃
J
z
s
r
,
t
s
r
,
t
+
1
) exp(
ih
L
r
∑
r
=
1
s
r
,
t
) exp(
iJ
z
L
r
∑
r
∈
even
s
r
,
t
s
r
+
1
,
t
)
=
k
L
r
L
t
L
t
∏
t
=
1
L
r
∏
r
=
1
exp(
iJ
z
s
r
,
t
+
1
s
r
+
1
,
t
+
1
) exp(
i
̃
J
z
s
r
,
t
s
r
,
t
+
1
) exp(
ihs
r
,
t
)
=
k
L
̃
t
L
̃
r
L
̃
t
∏
̃
t
=
1
L
̃
r
∏
̃
r
=
1
exp(
iJ
z
̃
s
̃
r
+
1
,
̃
t
̃
s
̃
r
+
1
,
̃
t
+
1
) exp(
i
̃
J
z
̃
s
̃
r
,
̃
t
̃
s
̃
r
+
1
,
̃
t
) exp(
ih
̃
s
̃
r
,
̃
t
)
=
k
L
̃
t
L
̃
r
L
̃
t
∏
̃
t
∈
odd
exp(
iJ
z
L
̃
r
∑
̃
r
∈
odd
̃
s
̃
r
+
1
,
̃
t
̃
s
̃
r
+
1
,
̃
t
+
1
) exp(
i
̃
J
z
L
̃
r
∑
̃
r
=
1
̃
s
̃
r
,
̃
t
̃
s
̃
r
+
1
,
̃
t
) exp(
ih
L
̃
r
∑
̃
r
=
1
̃
s
̃
r
,
̃
t
) exp(
iJ
z
L
̃
r
∑
̃
r
∈
odd
̃
s
̃
r
,
̃
t
̃
s
̃
r
,
̃
t
+
1
)
k
L
̃
t
L
̃
r
L
̃
t
∏
̃
t
∈
even
exp(
iJ
z
L
̃
r
∑
̃
r
∈
even
̃
s
̃
r
+
1
,
̃
t
̃
s
̃
r
+
1
,
̃
t
+
1
) exp(
i
̃
J
z
L
̃
r
∑
̃
r
=
1
̃
s
̃
r
,
̃
t
̃
s
̃
r
+
1
,
̃
t
) exp(
ih
L
̃
r
∑
̃
r
=
1
̃
s
̃
r
,
̃
t
) exp(
iJ
z
L
̃
r
∑
̃
r
∈
even
̃
s
̃
r
,
̃
t
̃
s
̃
r
,
̃
t
+
1
)
=
Tr
L
̃
r
(
ˆ
V
L
̃
t
)
(11)
with
ˆ
V
defined as
ˆ
V
=
(
kk
′
)
L
̃
r
exp(
i
∑
L
̃
r
̃
r
∈
odd
̃
J
z
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
) exp(
i
̃
J
x
∑
L
̃
r
̃
r
=
1
ˆ
σ
x
̃
r
) exp(
ih
∑
L
̃
r
̃
r
=
1
ˆ
σ
z
̃
r
+
i
̃
J
z
∑
L
̃
r
̃
r
∈
odd
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
)
t
=
odd
.
(
kk
′
)
L
̃
r
exp(
i
∑
L
̃
r
̃
r
∈
even
̃
J
z
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
) exp(
i
̃
J
x
∑
L
̃
r
̃
r
=
1
ˆ
σ
x
̃
r
) exp(
ih
∑
L
̃
r
̃
r
=
1
ˆ
σ
z
̃
r
+
i
̃
J
z
∑
L
̃
r
̃
r
∈
even
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
)
t
=
even
.
(12)
where
L
t
=
L
̃
r
,
L
r
=
L
̃
t
. Additionally, we emphasize the significance of the PBC, since the spatially
periodic boundary condition is exactly matched with the trace operation along the temporal direction after
the space-time duality.
III. THE SPACE-TIME DUALITY OF CORRELATION FUNCTION
F
(
φ
)
A. Non-Hermitian system via the space-time duality
We investigate the space-time duality of the correlation function
F
(
φ
) whose second-order derivative of
φ
is the out-of-time-ordered commutator(OTOC).
F
(
φ
) is defined as
F
(
φ
)
=
1
2
L
r
Tr
L
r
[
ˆ
O
(
L
t
)
e
i
φ
ˆ
W
ˆ
O
†
(
L
t
)
e
−
i
φ
ˆ
W
]
.
(13)
In our numerical result, we choose
ˆ
W
=
∑
r
ˆ
σ
z
r
as an operator that uniformly acts on all spatial sites, and
ˆ
O
=
ˆ
σ
z
1
which is a spatial local operator. Additionally, after writing the operator in the Heisenberg picture
5
(a)
V
^
L
t
1
1+L
t
1+2L
t
1+3L
t
FIG. 2: The double Keldysh contour demonstrated in the Fig. 2(b) of the main text but with numbering. Squares and
triangles label
e
±
i
φ
ˆ
W
and
ˆ
O
acting on di
ff
erent spatial points of the spatial contour in the dual circuit respectively. The
distance between the triangle and square is
L
t
. Boundary terms
e
±
i
φ
ˆ
W
are labeled by 1 and 2
L
t
+
1 separately, whereas
two local operators
ˆ
O
are labeled by
L
t
+
1 and 3
L
t
+
1.
explicitly, we have
F
(
φ
)
=
1
2
L
r
Tr
L
r
[
(
ˆ
U
)
†
L
t
ˆ
σ
z
1
(
ˆ
U
)
L
t
e
i
φ
∑
r
ˆ
σ
z
r
(
ˆ
U
)
†
L
t
ˆ
σ
z
1
(
ˆ
U
)
L
t
e
−
i
φ
∑
r
ˆ
σ
z
r
]
.
(14)
Following the same procedure in sec. I, we expand the
F
(
φ
) in the
ˆ
σ
z
diagonal basis
|
s
r
,
t
〉
F
(
φ
)
=
1
2
L
r
k
4
L
r
L
t
s
L
t
+
1
,
1
s
3
L
t
+
1
,
1
L
r
∏
r
=
1
exp(
i
4
L
t
∑
t
=
1
J
z
,
t
s
r
,
t
s
r
+
1
,
t
) exp(
i
4
L
t
∑
t
=
1
̃
J
z
,
t
s
r
,
t
s
r
,
t
+
1
) exp(
i
4
L
t
∑
t
=
1
̃
h
t
s
r
,
t
)
e
−
i
φ
s
r
,
1
e
i
φ
s
r
,
2
L
t
+
1
.
(15)
The labeling is according to the convention illustrated in the Fig. 2. Furthermore,
F
(
φ
) can be represented
in terms of the new classical variable ̃
s
:
F
(
φ
)
=
1
2
L
̃
t
k
L
̃
t
L
̃
r
̃
s
1
,
L
t
+
1
̃
s
1
,
3
L
t
+
1
L
̃
t
∏
̃
t
=
1
exp(
i
L
̃
r
∑
̃
r
=
1
J
z
,
̃
r
̃
s
̃
r
,
̃
t
̃
s
̃
r
,
̃
t
+
1
) exp(
i
L
̃
r
∑
̃
r
=
1
̃
J
z
,
̃
r
̃
s
̃
r
,
̃
t
̃
s
̃
r
+
1
,
̃
t
) exp(
i
L
̃
r
∑
̃
r
=
1
̃
h
̃
r
̃
s
̃
r
,
̃
t
)
e
−
i
φ
̃
s
1
,
̃
t
e
i
φ
̃
s
2
L
t
+
1
,
̃
t
,
(16)
where 4
L
t
=
L
̃
r
,
L
r
=
L
̃
t
, and the parameters are shown in the Eq. (20). Transforming it into the trace
formula, we have
F
(
φ
)
=
Tr
L
̃
r
[
(
ˆ
V
I
ˆ
B
(
φ
))
L
̃
t
ˆ
̃
O
L
t
+
1
ˆ
̃
O
3
L
t
+
1
]
Tr
L
̃
r
[
(
ˆ
V
I
ˆ
B
(
φ
))
L
̃
t
]
,
(17)
with
ˆ
V
I
=
1
2
L
̃
t
(
kk
′
)
L
̃
t
L
̃
r
3
∏
j
=
0
(
ˆ
σ
0
1
+
jL
t
+
ˆ
σ
x
1
+
jL
t
)
exp
i
L
̃
r
∑
̃
r
=
1
̃
J
x
,
̃
r
ˆ
σ
x
̃
r
exp
i
L
̃
r
∑
̃
r
=
1
̃
J
z
,
̃
r
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
+
i
L
̃
r
∑
̃
r
=
1
̃
h
̃
r
ˆ
σ
z
̃
r
,
(18)
and
ˆ
B
(
φ
)
=
e
i
φ
ˆ
σ
z
2
L
t
+
1
e
−
i
φ
ˆ
σ
z
1
.
(19)
6
The parameters on the contour are
J
z
,
̃
r
̃
J
x
,
̃
r
̃
J
z
,
̃
r
̃
h
̃
r
̃
r
=
1
0
0
̃
J
z
0
1
<
̃
r
<
L
t
+
1
J
z
̃
J
x
̃
J
z
h
̃
r
=
L
t
+
1
0
0
̃
J
′
z
0
L
t
+
1
<
̃
r
<
2
L
t
+
1
−
J
z
̃
J
′
x
̃
J
′
z
−
h
̃
r
=
2
L
t
+
1
0
0
̃
J
z
0
2
L
t
+
1
<
̃
r
<
3
L
t
+
1
J
z
̃
J
x
̃
J
z
h
̃
r
=
3
L
t
+
1
0
0
̃
J
′
z
0
3
L
t
+
1
<
̃
r
<
4
L
t
+
1
−
J
z
̃
J
′
x
̃
J
′
z
−
h
with
̃
J
x
=
arctan(
−
i
exp(
−
2
iJ
z
))
̃
J
′
x
=
arctan(
−
i
exp(2
iJ
z
))
=
̃
J
x
+
π/
2
̃
J
z
=
−
π/
4
+
i
2
ln(tan
J
x
)
̃
J
′
z
=
−
π/
4
+
i
2
ln(tan(
−
J
x
))
=
̃
J
z
−
π/
2
.
(20)
From the Fig. 2 and the Eq. (20), we notice that the parameters of the qubits at ̃
r
=
1
,
L
t
+
1
,
2
L
t
+
1
,
3
L
t
+
1
are special, since these sites connect the forwards and backwards evolutions. We call these qubits
edge
qubits
for later convenience. The Eq. (18) shows, projectors ( ˆ
σ
0
̃
t
+
ˆ
σ
x
̃
t
) exist at each edge qubit, since the
J
z
,
̃
r
=
0 at these sites. We will give a proof in the sec. IV.
Finally, we note that
ˆ
V
in general describes a non-unitary evolution whereas
ˆ
U
is unitary. Firstly, pa-
rameters
{
̃
J
x
,
̃
J
z
,
̃
J
′
x
,
̃
J
′
z
}
in
ˆ
V
have non-vanishing imaginary part, but parameters in
ˆ
U
are all real numbers.
Secondly, the projectors in Eq. (18) lead to intrinsically non-unitary evolution.
B. Hermitian system
To provide further evidence on the non-Hermitian boundary e
ff
ect, we artificially change the parameters
̃
J
x
,
̃
J
z
and
̃
h
in
ˆ
V
to be purely imaginary and throw away the projectors at edge qubits, such that
ˆ
V
behaves
as
e
−
ˆ
H
in which
ˆ
H
is a Hermitian operator. Explicitly, the correlator
F
(
φ
) can be formulated as
F
(
φ
)
=
Tr
L
̃
r
[
(
ˆ
V
II
ˆ
B
(
φ
))
L
̃
t
ˆ
̃
O
L
t
+
1
ˆ
̃
O
3
L
t
+
1
]
Tr
L
̃
r
[
(
ˆ
V
II
ˆ
B
(
φ
))
L
̃
t
]
,
(21)
with
ˆ
V
II
=
exp
−
β
L
̃
r
∑
̃
r
=
1
̃
J
x
,
̃
t
ˆ
σ
x
̃
r
exp
−
β
L
̃
r
∑
̃
r
=
1
̃
J
z
,
̃
t
ˆ
σ
z
̃
r
ˆ
σ
z
̃
r
+
1
−
β
L
̃
r
∑
̃
r
=
1
̃
h
̃
t
ˆ
σ
z
̃
r
,
(22)
7
where
β
is the inverse temperature and
ˆ
B
(
φ
) is defined in Eq. (18). We summarize the parameters on the
contour in the following table
̃
J
x
,
̃
r
̃
J
z
,
̃
r
̃
h
̃
r
̃
r
=
1
J
x
J
z
h
1
<
̃
r
<
L
t
+
1
J
x
J
z
h
̃
r
=
L
t
+
1
J
x
J
z
h
L
t
+
1
<
̃
r
<
2
L
t
+
1
J
x
J
z
h
̃
r
=
2
L
t
+
1
J
x
J
z
h
2
L
t
+
1
<
̃
r
<
3
L
t
+
1
J
x
J
z
h
̃
r
=
3
L
t
+
1
J
x
J
z
h
3
L
t
+
1
<
̃
r
<
4
L
t
+
1
J
x
J
z
h
(23)
In our numerical calculation, we set
J
z
=
J
x
=
1
,
h
=
0
.
5
,β
=
0
.
1. In the high-temperature limit, the
Eq. (22) approximately acts as the thermal density matrix average with Hermitian Hamiltonian
ˆ
H
.
IV. FORCE PROJECTIVE MEASUREMENT ON THE EDGE QUBITS
We discuss the details of the force projective measurement at edge qubits(defined at the sec. III) in the
Eq. (18). First of all, we prove a proposition on a minimal case.
Proposition
When
J
z
=
0, the two-qubit gate exp(
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
) on the spatial sites [
r
,
r
+
1] is mapped to
a single-qubit gate ( ˆ
σ
0
+
ˆ
σ
z
)
t
on the temporal site
t
.
Proof.
We consider a general two-qubit gate ˆ
u
o
1
,
o
2
i
1
,
i
2
, where
i
1
,
i
2
denote two input qubits and
o
1
,
o
2
denote
two output qubits. Via the space-time duality, this gate is dual to another two-qubit gate, with
ˆ
̃
u
o
1
,
o
2
i
1
,
i
2
≡
ˆ
u
i
2
,
o
2
i
1
,
o
1
.
(24)
Particularly, for the two-qubit gate exp
(
iJ
z
ˆ
σ
z
r
ˆ
σ
z
r
+
1
)
, we compute the matrix element in the ˆ
σ
z
eigen-basis
(
|−
1
,
−
1
〉
,
|−
1
,
1
〉
,
|
1
,
−
1
〉
,
|
1
,
1
〉
)
ˆ
u
=
exp(
iJ
z
)
0
0
0
0
exp(
−
iJ
z
)
0
0
0
0
exp(
−
iJ
z
)
0
0
0
0
exp(
iJ
z
)
.
(25)
8
After the space-time duality, we get
ˆ
̃
u
=
exp(
iJ
z
) 0 0 exp(
−
iJ
z
)
0
0 0
0
0
0 0
0
exp(
−
iJ
z
) 0 0 exp(
iJ
z
)
.
(26)
Although ̃
u
is originally defined as a 4-dimensional matrix, it is easy to find that ̃
u
has a 2-dimensional
irreducible representation whose basis are
|−
1
,
−
1
〉
and
|
1
,
1
〉
. Thus, this two-qubit gate ̃
u
can be regarded
as a single-qubit gate on time
t
, in the reduced basis (
|−
1
,
−
1
〉
,
|
1
,
1
〉
) with
ˆ
̃
u
=
exp(
iJ
z
) exp(
−
iJ
z
)
exp(
−
iJ
z
) exp(
iJ
z
)
.
(27)
In the case of
J
z
=
0,
ˆ
̃
u
∣
∣
∣
∣
J
z
=
0
=
1 1
1 1
=
ˆ
σ
0
+
ˆ
σ
x
.
(28)
This completes the proof of the proposition.
Secondly, at four edge qubits, the backwards evolution
ˆ
U
is adjacent to the forwards evolution
ˆ
U
†
. It
can be verified that these four cases
•
(
ˆ
U
†
) ˆ
σ
z
1
(
ˆ
U
)
•
(
ˆ
U
)
e
i
φ
∑
r
ˆ
σ
z
r
(
ˆ
U
)
†
•
(
ˆ
U
†
) ˆ
σ
z
1
(
ˆ
U
)
•
(
ˆ
U
)
e
−
i
φ
∑
r
ˆ
σ
z
r
(
ˆ
U
)
†
all lead to
J
z
,
̃
r
=
0 in the Eq. (16) e
ff
ectively. Applying the proposition above, one can immediately obtain
the force projective measurement
[
∏
3
j
=
0
(
ˆ
σ
0
1
+
jL
t
+
ˆ
σ
x
1
+
jL
t
)]
in the Eq. (18).
∗
They contribute equally to this work.
†
PengfeiZhang.physics@gmail.com
‡
hzhai@mail.tsinghua.edu.cn