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, m
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 NumKet, Z
from proveit.physics.quantum.QPE import _Psi_ket, _ket_u, _s, _s_wire, _t
from proveit.physics.quantum.circuits import Input, Measure, 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)
expr = Equals(Len(operands = [ExprRange(sub_expr1, MultiQubitElem(element = Input(state = TensorProd(_Psi_ket, _ket_u), part = sub_expr1), targets = Interval(one, sub_expr3)), one, sub_expr3), ExprRange(sub_expr1, Measure(basis = Z), one, _t), _s_wire, ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(m, _t), part = sub_expr1), targets = Interval(one, _t)), 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 \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 \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 \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{
& & \meter 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \end{array}, ..\left(t - 3\right) \times.., \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \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 m \rangle_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\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: 72 end_index: 61
9	ExprRange	lambda_map: 15 start_index: 72 end_index: 73
10	ExprRange	lambda_map: 16 start_index: 72 end_index: 74
11	ExprRange	lambda_map: 17 start_index: 72 end_index: 73
12	ExprRange	lambda_map: 18 start_index: 72 end_index: 74
13	ExprRange	lambda_map: 19 start_index: 72 end_index: 20
14	Lambda	parameter: 59 body: 21
15	Lambda	parameter: 59 body: 22
16	Lambda	parameter: 59 body: 23
17	Lambda	parameter: 59 body: 24
18	Lambda	parameter: 59 body: 25
19	Lambda	parameter: 56 body: 27
20	Literal
21	Operation	operator: 34 operands: 28
22	Operation	operator: 29 operands: 30
23	Operation	operator: 31 operands: 32
24	Operation	operator: 34 operands: 33
25	Operation	operator: 34 operands: 35
26	ExprTuple	56
27	ExprRange	lambda_map: 36 start_index: 72 end_index: 61
28	NamedExprs	element: 37 targets: 38
29	Literal
30	NamedExprs	basis: 39
31	Literal
32	NamedExprs	operation: 40
33	NamedExprs	element: 41 targets: 42
34	Literal
35	NamedExprs	element: 43 targets: 44
36	Lambda	parameter: 59 body: 46
37	Operation	operator: 47 operands: 48
38	Operation	operator: 54 operands: 49
39	Literal
40	Literal
41	Operation	operator: 52 operands: 50
42	Operation	operator: 54 operands: 51
43	Operation	operator: 52 operands: 53
44	Operation	operator: 54 operands: 55
45	ExprTuple	59
46	ExprTuple	56, 59
47	Literal
48	NamedExprs	state: 57 part: 59
49	ExprTuple	72, 61
50	NamedExprs	state: 58 part: 59
51	ExprTuple	72, 73
52	Literal
53	NamedExprs	state: 70 part: 59
54	Literal
55	ExprTuple	60, 61
56	Variable
57	Operation	operator: 62 operands: 63
58	Operation	operator: 64 operands: 65
59	Variable
60	Operation	operator: 67 operands: 66
61	Operation	operator: 67 operands: 68
62	Literal
63	ExprTuple	69, 70
64	Literal
65	ExprTuple	71, 73
66	ExprTuple	73, 72
67	Literal
68	ExprTuple	73, 74
69	Literal
70	Literal
71	Variable
72	Literal
73	Literal
74	Literal