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 _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = frac(one, sqrt(two))
sub_expr3 = ExprRange(sub_expr1, ScalarMult(sub_expr2, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), Add(Neg(t), one), zero).with_decreasing_order()
expr = Equals(ScalarMult(frac(one, Exp(two, subtract(frac(Add(t, one), two), frac(t, two)))), TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), sub_expr3)), ScalarMult(sub_expr2, TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, two_pow_t, _phase)), ket1)), sub_expr3)))

# 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) = \left(\frac{1}{\sqrt{2}} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t} \cdot \varphi} \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)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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')

	core type	sub-expressions
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 61 operands: 5
4	Operation	operator: 61 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	37, 9
7	Operation	operator: 69 operands: 10
8	Operation	operator: 12 operands: 11
9	Operation	operator: 12 operands: 13
10	ExprTuple	77, 14
11	ExprTuple	15, 17
12	Literal
13	ExprTuple	16, 17
14	Operation	operator: 84 operands: 18
15	Operation	operator: 45 operands: 19
16	Operation	operator: 45 operands: 20
17	ExprRange	lambda_map: 21 start_index: 22 end_index: 65
18	ExprTuple	86, 23
19	ExprTuple	52, 24
20	ExprTuple	52, 25
21	Lambda	parameter: 90 body: 26
22	Operation	operator: 54 operands: 27
23	Operation	operator: 54 operands: 28
24	Operation	operator: 61 operands: 29
25	Operation	operator: 61 operands: 30
26	Operation	operator: 61 operands: 31
27	ExprTuple	32, 77
28	ExprTuple	33, 34
29	ExprTuple	35, 67
30	ExprTuple	36, 67
31	ExprTuple	37, 38
32	Operation	operator: 88 operand: 74
33	Operation	operator: 69 operands: 40
34	Operation	operator: 88 operand: 48
35	Operation	operator: 84 operands: 42
36	Operation	operator: 84 operands: 43
37	Operation	operator: 69 operands: 44
38	Operation	operator: 45 operands: 46
39	ExprTuple	74
40	ExprTuple	47, 86
41	ExprTuple	48
42	ExprTuple	75, 49
43	ExprTuple	75, 50
44	ExprTuple	77, 51
45	Literal
46	ExprTuple	52, 53
47	Operation	operator: 54 operands: 55
48	Operation	operator: 69 operands: 56
49	Operation	operator: 78 operands: 57
50	Operation	operator: 78 operands: 58
51	Operation	operator: 84 operands: 59
52	Operation	operator: 72 operand: 65
53	Operation	operator: 61 operands: 62
54	Literal
55	ExprTuple	74, 77
56	ExprTuple	74, 86
57	ExprTuple	86, 80, 81, 83, 63
58	ExprTuple	86, 80, 81, 63, 83
59	ExprTuple	86, 64
60	ExprTuple	65
61	Literal
62	ExprTuple	66, 67
63	Operation	operator: 84 operands: 68
64	Operation	operator: 69 operands: 70
65	Literal
66	Operation	operator: 84 operands: 71
67	Operation	operator: 72 operand: 77
68	ExprTuple	86, 74
69	Literal
70	ExprTuple	77, 86
71	ExprTuple	75, 76
72	Literal
73	ExprTuple	77
74	Variable
75	Literal
76	Operation	operator: 78 operands: 79
77	Literal
78	Literal
79	ExprTuple	86, 80, 81, 82, 83
80	Literal
81	Literal
82	Operation	operator: 84 operands: 85
83	Literal
84	Literal
85	ExprTuple	86, 87
86	Literal
87	Operation	operator: 88 operand: 90
88	Literal
89	ExprTuple	90
90	Variable

Expression of type Equals¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Equals ¶

from the theory of proveit.physics.quantum.QPE ¶