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, Lambda, 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, five, one
from proveit.physics.quantum import ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _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 = Add(_t, _s)
sub_expr3 = Interval(one, sub_expr2)
sub_expr4 = Lambda(sub_expr1, MultiQubitElem(element = Gate(operation = QPE1(_U, _t), part = sub_expr1), targets = sub_expr3))
expr = Equals(Len(operands = [Lambda(sub_expr1, Input(state = ket_plus)), Lambda(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = Interval(Add(_t, one), sub_expr2))), sub_expr4, sub_expr4, Lambda(sub_expr1, MultiQubitElem(element = Output(state = TensorProd(_psi__t_ket, _ket_u), part = sub_expr1), targets = sub_expr3))]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, five)]))

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({_{-}a} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert + \rangle} & \qw 
} \end{array}, {_{-}a} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert u \rangle~\mbox{part}~{_{-}a}~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, {_{-}a} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}_1\left(U, t\right)~\mbox{part}~{_{-}a}~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, {_{-}a} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}_1\left(U, t\right)~\mbox{part}~{_{-}a}~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, {_{-}a} \mapsto \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~{_{-}a}~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} 
} \end{array}\right)| = |\left(1, 2, \ldots, 5\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, 10, 11
6	Literal
7	ExprTuple	12
8	Lambda	parameter: 46 body: 13
9	Lambda	parameter: 46 body: 14
10	Lambda	parameter: 46 body: 15
11	Lambda	parameter: 46 body: 16
12	ExprRange	lambda_map: 17 start_index: 55 end_index: 18
13	Operation	operator: 33 operands: 19
14	Operation	operator: 22 operands: 20
15	Operation	operator: 22 operands: 21
16	Operation	operator: 22 operands: 23
17	Lambda	parameter: 46 body: 46
18	Literal
19	NamedExprs	state: 25
20	NamedExprs	element: 26 targets: 27
21	NamedExprs	element: 28 targets: 30
22	Literal
23	NamedExprs	element: 29 targets: 30
24	ExprTuple	46
25	Operation	operator: 31 operand: 42
26	Operation	operator: 33 operands: 34
27	Operation	operator: 40 operands: 35
28	Operation	operator: 36 operands: 37
29	Operation	operator: 38 operands: 39
30	Operation	operator: 40 operands: 41
31	Literal
32	ExprTuple	42
33	Literal
34	NamedExprs	state: 58 part: 46
35	ExprTuple	43, 47
36	Literal
37	NamedExprs	operation: 44 part: 46
38	Literal
39	NamedExprs	state: 45 part: 46
40	Literal
41	ExprTuple	55, 47
42	Literal
43	Operation	operator: 53 operands: 48
44	Operation	operator: 49 operands: 50
45	Operation	operator: 51 operands: 52
46	Variable
47	Operation	operator: 53 operands: 54
48	ExprTuple	62, 55
49	Literal
50	ExprTuple	56, 62
51	Literal
52	ExprTuple	57, 58
53	Literal
54	ExprTuple	62, 59
55	Literal
56	Literal
57	Operation	operator: 60 operand: 62
58	Literal
59	Literal
60	Literal
61	ExprTuple	62
62	Literal