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, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, subtract, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import SubIndexed, _phase, _psi, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(t, one)
expr = Equals(ScalarMult(frac(one, Exp(two, subtract(frac(sub_expr2, two), frac(t, two)))), TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), ExprRange(sub_expr1, ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), Add(Neg(t), one), zero).with_decreasing_order())), SubIndexed(_psi, [sub_expr2]))

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(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right){\otimes} \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) {\otimes}  \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) {\otimes}  \ldots {\otimes}  \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right)\right)\right) = \lvert \psi_{t + 1} \rangle

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: 52 operands: 5
4	Operation	operator: 6 operand: 40
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	40
8	Operation	operator: 60 operands: 10
9	Operation	operator: 11 operands: 12
10	ExprTuple	68, 13
11	Literal
12	ExprTuple	14, 15
13	Operation	operator: 75 operands: 16
14	Operation	operator: 38 operands: 17
15	ExprRange	lambda_map: 18 start_index: 19 end_index: 56
16	ExprTuple	77, 20
17	ExprTuple	44, 21
18	Lambda	parameter: 81 body: 22
19	Operation	operator: 46 operands: 23
20	Operation	operator: 46 operands: 24
21	Operation	operator: 52 operands: 25
22	Operation	operator: 52 operands: 26
23	ExprTuple	27, 68
24	ExprTuple	28, 29
25	ExprTuple	30, 58
26	ExprTuple	31, 32
27	Operation	operator: 79 operand: 65
28	Operation	operator: 60 operands: 34
29	Operation	operator: 79 operand: 41
30	Operation	operator: 75 operands: 36
31	Operation	operator: 60 operands: 37
32	Operation	operator: 38 operands: 39
33	ExprTuple	65
34	ExprTuple	40, 77
35	ExprTuple	41
36	ExprTuple	66, 42
37	ExprTuple	68, 43
38	Literal
39	ExprTuple	44, 45
40	Operation	operator: 46 operands: 47
41	Operation	operator: 60 operands: 48
42	Operation	operator: 69 operands: 49
43	Operation	operator: 75 operands: 50
44	Operation	operator: 63 operand: 56
45	Operation	operator: 52 operands: 53
46	Literal
47	ExprTuple	65, 68
48	ExprTuple	65, 77
49	ExprTuple	77, 71, 72, 74, 54
50	ExprTuple	77, 55
51	ExprTuple	56
52	Literal
53	ExprTuple	57, 58
54	Operation	operator: 75 operands: 59
55	Operation	operator: 60 operands: 61
56	Literal
57	Operation	operator: 75 operands: 62
58	Operation	operator: 63 operand: 68
59	ExprTuple	77, 65
60	Literal
61	ExprTuple	68, 77
62	ExprTuple	66, 67
63	Literal
64	ExprTuple	68
65	Variable
66	Literal
67	Operation	operator: 69 operands: 70
68	Literal
69	Literal
70	ExprTuple	77, 71, 72, 73, 74
71	Literal
72	Literal
73	Operation	operator: 75 operands: 76
74	Literal
75	Literal
76	ExprTuple	77, 78
77	Literal
78	Operation	operator: 79 operand: 81
79	Literal
80	ExprTuple	81
81	Variable