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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.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, frac, i, one, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _alpha_m, _m_domain, _phase, _t
In [2]:
# build up the expression from sub-expressions
expr = Equals(_alpha_m, Mult(frac(one, Exp(two, frac(_t, two))), Qmult(NumBra(m, _t), InverseFourierTransform(_t), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), 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} = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \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)
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: 25
4Operationoperator: 49
operands: 7
5Literal
6ExprTuple25
7ExprTuple8, 9
8Operationoperator: 27
operands: 10
9Operationoperator: 11
operands: 12
10ExprTuple65, 13
11Literal
12ExprTuple14, 15, 16
13Operationoperator: 59
operands: 17
14Operationoperator: 18
operands: 19
15Operationoperator: 20
operand: 64
16Operationoperator: 22
operand: 26
17ExprTuple63, 24
18Literal
19ExprTuple25, 64
20Literal
21ExprTuple64
22Literal
23ExprTuple26
24Operationoperator: 27
operands: 28
25Variable
26Lambdaparameter: 56
body: 30
27Literal
28ExprTuple64, 63
29ExprTuple56
30Conditionalvalue: 31
condition: 32
31Operationoperator: 33
operands: 34
32Operationoperator: 35
operands: 36
33Literal
34ExprTuple37, 38
35Literal
36ExprTuple56, 39
37Operationoperator: 59
operands: 40
38Operationoperator: 41
operands: 42
39Operationoperator: 43
operands: 44
40ExprTuple45, 46
41Literal
42ExprTuple56, 64
43Literal
44ExprTuple47, 48
45Literal
46Operationoperator: 49
operands: 50
47Literal
48Operationoperator: 51
operands: 52
49Literal
50ExprTuple63, 53, 54, 55, 56
51Literal
52ExprTuple57, 58
53Literal
54Literal
55Literal
56Variable
57Operationoperator: 59
operands: 60
58Operationoperator: 61
operand: 65
59Literal
60ExprTuple63, 64
61Literal
62ExprTuple65
63Literal
64Literal
65Literal