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 k, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Add(t, one)
sub_expr3 = frac(t, two)
sub_expr4 = frac(sub_expr2, two)
sub_expr5 = Exp(e, Mult(two, pi, i, _phase, k))
expr = Equals(ScalarMult(frac(one, Exp(two, sub_expr4)), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr5, NumKet(k, sub_expr2)), domain = Interval(zero, subtract(Mult(two, two_pow_t), one)))), TensorProd(ScalarMult(frac(one, Exp(two, subtract(sub_expr4, sub_expr3))), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1))), ScalarMult(frac(one, Exp(two, sub_expr3)), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr5, NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one))))))

# 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}}} \cdot \left(\sum_{k=0}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right)\right) = \left(\left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right)\right) {\otimes} \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\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: 57 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10, 11
8	Operation	operator: 86 operands: 12
9	Operation	operator: 28 operand: 17
10	Operation	operator: 57 operands: 14
11	Operation	operator: 57 operands: 15
12	ExprTuple	104, 16
13	ExprTuple	17
14	ExprTuple	18, 19
15	ExprTuple	20, 21
16	Operation	operator: 98 operands: 22
17	Lambda	parameter: 95 body: 23
18	Operation	operator: 86 operands: 24
19	Operation	operator: 25 operands: 26
20	Operation	operator: 86 operands: 27
21	Operation	operator: 28 operand: 36
22	ExprTuple	102, 62
23	Conditional	value: 30 condition: 31
24	ExprTuple	104, 32
25	Literal
26	ExprTuple	33, 34
27	ExprTuple	104, 35
28	Literal
29	ExprTuple	36
30	Operation	operator: 57 operands: 37
31	Operation	operator: 59 operands: 38
32	Operation	operator: 98 operands: 39
33	Operation	operator: 56 operand: 82
34	Operation	operator: 57 operands: 41
35	Operation	operator: 98 operands: 42
36	Lambda	parameter: 95 body: 44
37	ExprTuple	65, 45
38	ExprTuple	95, 46
39	ExprTuple	102, 47
40	ExprTuple	82
41	ExprTuple	48, 49
42	ExprTuple	102, 79
43	ExprTuple	95
44	Conditional	value: 50 condition: 51
45	Operation	operator: 73 operands: 52
46	Operation	operator: 75 operands: 53
47	Operation	operator: 90 operands: 54
48	Operation	operator: 98 operands: 55
49	Operation	operator: 56 operand: 104
50	Operation	operator: 57 operands: 58
51	Operation	operator: 59 operands: 60
52	ExprTuple	95, 78
53	ExprTuple	82, 61
54	ExprTuple	62, 63
55	ExprTuple	80, 64
56	Literal
57	Literal
58	ExprTuple	65, 66
59	Literal
60	ExprTuple	95, 67
61	Operation	operator: 90 operands: 68
62	Operation	operator: 86 operands: 69
63	Operation	operator: 100 operand: 79
64	Operation	operator: 88 operands: 71
65	Operation	operator: 98 operands: 72
66	Operation	operator: 73 operands: 74
67	Operation	operator: 75 operands: 76
68	ExprTuple	77, 97
69	ExprTuple	78, 102
70	ExprTuple	79
71	ExprTuple	102, 92, 93, 94, 96
72	ExprTuple	80, 81
73	Literal
74	ExprTuple	95, 103
75	Literal
76	ExprTuple	82, 83
77	Operation	operator: 88 operands: 84
78	Operation	operator: 90 operands: 85
79	Operation	operator: 86 operands: 87
80	Literal
81	Operation	operator: 88 operands: 89
82	Literal
83	Operation	operator: 90 operands: 91
84	ExprTuple	102, 96
85	ExprTuple	103, 104
86	Literal
87	ExprTuple	103, 102
88	Literal
89	ExprTuple	102, 92, 93, 94, 95
90	Literal
91	ExprTuple	96, 97
92	Literal
93	Literal
94	Literal
95	Variable
96	Operation	operator: 98 operands: 99
97	Operation	operator: 100 operand: 104
98	Literal
99	ExprTuple	102, 103
100	Literal
101	ExprTuple	104
102	Literal
103	Variable
104	Literal

Expression of type Equals¶

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

Expression of type Equals ¶

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