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Expression of type Equals

from the theory of proveit.physics.quantum.QPE

In [1]:
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, m
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Neg, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _alpha_m_mod_two_pow_t, _m_domain, _phase, _two_pow_t
In [2]:
# build up the expression from sub-expressions
expr = Equals(_alpha_m_mod_two_pow_t, Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), domain = _m_domain)))
expr:
In [3]:
# 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
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
\alpha_{m ~\textup{mod}~ 2^{t}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right)\right)\right)
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(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')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 5
operand: 8
4Operationoperator: 50
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 10
8Operationoperator: 11
operands: 12
9Operationoperator: 43
operands: 13
10Operationoperator: 14
operand: 16
11Literal
12ExprTuple57, 48
13ExprTuple49, 48
14Literal
15ExprTuple16
16Lambdaparameter: 56
body: 18
17ExprTuple56
18Conditionalvalue: 19
condition: 20
19Operationoperator: 50
operands: 21
20Operationoperator: 22
operands: 23
21ExprTuple24, 25
22Literal
23ExprTuple56, 26
24Operationoperator: 52
operands: 27
25Operationoperator: 52
operands: 28
26Operationoperator: 29
operands: 30
27ExprTuple32, 31
28ExprTuple32, 33
29Literal
30ExprTuple34, 35
31Operationoperator: 45
operand: 40
32Literal
33Operationoperator: 50
operands: 37
34Literal
35Operationoperator: 38
operands: 39
36ExprTuple40
37ExprTuple58, 54, 55, 41, 56
38Literal
39ExprTuple48, 42
40Operationoperator: 43
operands: 44
41Literal
42Operationoperator: 45
operand: 49
43Literal
44ExprTuple47, 48
45Literal
46ExprTuple49
47Operationoperator: 50
operands: 51
48Operationoperator: 52
operands: 53
49Literal
50Literal
51ExprTuple58, 54, 55, 56, 57
52Literal
53ExprTuple58, 59
54Literal
55Literal
56Variable
57Variable
58Literal
59Literal