Expression of type Equals

from the theory of proveit.physics.quantum.QPE

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import ExprRange, Variable
from proveit.core_expr_types import Len
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals
from proveit.numbers import Add, Interval, one, three
from proveit.physics.quantum import I
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _Psi_ket, _ket_u, _psi__t_ket, _s, _t
from proveit.physics.quantum.circuits import Gate, Input, MultiQubitElem, Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr3 = Add(_t, _s)
sub_expr4 = Interval(one, _t)
expr = Equals(Len(operands = [ExprRange(sub_expr1, MultiQubitElem(element = Input(state = TensorProd(_psi__t_ket, _ket_u), part = sub_expr1), targets = Interval(one, sub_expr3)), one, sub_expr3), ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, _s), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = Interval(Add(_t, one), sub_expr3)), one, _s)]), Len(operands = [ExprRange(sub_expr2, ExprRange(sub_expr1, [sub_expr2, sub_expr1], one, sub_expr3), one, three)]))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

|\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~t + s~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, ..\left(s - 3\right) \times.., \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right)| = |\left(\left(1, 1\right), \left(1, 2\right), \ldots, \left(1, t + s\right), \left(2, 1\right), \left(2, 2\right), \ldots, \left(2, t + s\right), \ldots\ldots, \left(3, 1\right), \left(3, 2\right), \ldots, \left(3, t + s\right)\right)|

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9, 10, 11, 12
6	Literal
7	ExprTuple	13
8	ExprRange	lambda_map: 14 start_index: 71 end_index: 62
9	ExprRange	lambda_map: 15 start_index: 71 end_index: 75
10	ExprRange	lambda_map: 16 start_index: 71 end_index: 72
11	ExprRange	lambda_map: 17 start_index: 71 end_index: 75
12	ExprRange	lambda_map: 18 start_index: 71 end_index: 72
13	ExprRange	lambda_map: 19 start_index: 71 end_index: 20
14	Lambda	parameter: 60 body: 21
15	Lambda	parameter: 60 body: 22
16	Lambda	parameter: 60 body: 23
17	Lambda	parameter: 60 body: 24
18	Lambda	parameter: 60 body: 25
19	Lambda	parameter: 56 body: 27
20	Literal
21	Operation	operator: 32 operands: 28
22	Operation	operator: 32 operands: 29
23	Operation	operator: 48 operands: 30
24	Operation	operator: 32 operands: 31
25	Operation	operator: 32 operands: 33
26	ExprTuple	56
27	ExprRange	lambda_map: 34 start_index: 71 end_index: 62
28	NamedExprs	element: 35 targets: 36
29	NamedExprs	element: 37 targets: 40
30	NamedExprs	operation: 38
31	NamedExprs	element: 39 targets: 40
32	Literal
33	NamedExprs	element: 41 targets: 42
34	Lambda	parameter: 60 body: 44
35	Operation	operator: 45 operands: 46
36	Operation	operator: 54 operands: 47
37	Operation	operator: 48 operands: 49
38	Literal
39	Operation	operator: 52 operands: 50
40	Operation	operator: 54 operands: 51
41	Operation	operator: 52 operands: 53
42	Operation	operator: 54 operands: 55
43	ExprTuple	60
44	ExprTuple	56, 60
45	Literal
46	NamedExprs	state: 57 part: 60
47	ExprTuple	71, 62
48	Literal
49	NamedExprs	operation: 58 part: 60
50	NamedExprs	state: 59 part: 60
51	ExprTuple	71, 75
52	Literal
53	NamedExprs	state: 70 part: 60
54	Literal
55	ExprTuple	61, 62
56	Variable
57	Operation	operator: 63 operands: 64
58	Operation	operator: 65 operand: 75
59	Literal
60	Variable
61	Operation	operator: 67 operands: 66
62	Operation	operator: 67 operands: 68
63	Literal
64	ExprTuple	69, 70
65	Literal
66	ExprTuple	75, 71
67	Literal
68	ExprTuple	75, 72
69	Operation	operator: 73 operand: 75
70	Literal
71	Literal
72	Literal
73	Literal
74	ExprTuple	75
75	Literal