CC BY
6
DOI: 10.33693/2313-223X-2021-8-3-23-28
Jordan-Wigner Transformation and Qubits with Nontrivial Exchange Rule
A.Yu. Vlasov ©
P.V. Ramzaev Research Institute of Radiation Hygiene, Saint Petersburg, Russian Federation
E-mail: alex.yu.vlasov@gmail.com
Abstract. Well-known (spinless) fermionic qubits may need more subtle consideration in comparison with usual (spinful) fermions. Taking into account a model with local fermionic modes, formally only the 'occupied' states |1) could be relevant for antisymmetry with respect to particles interchange, but 'vacuum' state |0) is not. Introduction of exchange rule for such fermionic qubits indexed by some 'positions' may look questionable due to general super-selection principle. However, a consistent algebraic construction of such 'super-indexed' qubits is presented in this work. Considered method has some relation with construction of super-spaces, but it has some differences with standard definition of supersymmety sometimes used for generalizations of qubit model. Key words: Jordan-Wigner transformation, qubits with nontrivial exchange rule, Quantum Informatics
Acknowledgements. Author gratefully acknowledges possibility to present some topics considered here in a talk at Conference "Quantum Informatics - 2021," Moscow.
f _ —
FOR CITATION: Vlasov A.Yu. Jordan-Wigner Transformation and Qubits with Nontrivial Exchange Rule. Computational Nanotechnology. 2021. Vol. 8. No. 3. Pp. 23-28. DOI: 10.33693/2313-223X-2021-8-3-23-28
V J
DOI: 10.33693/2313-223X-2021-8-3-23-28
Преобразование Жордана-Вигнера
и кубиты с нетривиальными перестановочными соотношениями
А.Ю. Власов ©
Санкт-Петербургский научно-исследовательский институт радиационной гигиены имени профессора П.В. Рамзаева г. Санкт-Петербург, Российская Федерация
E-mail: alex.yu.vlasov@gmail.com
Аннотация. Хорошо известные фермионные кубиты (без спина) могут потребовать более тонкого рассмотрения по сравнению с обычными фермионами (со спином). Принимая во внимание модель с локальными фермионными модами, необходимо отметить, что формально только «занятые» состояния |1) могут подчиняться принципу антисимметрии по отношению к перестановкам, но это не относится к «вакуумным» состояниям |0). Введение перестановочных соотношений для таких фермионных кубитов, проиндексированных занимаемыми ими позициями, может вызвать определенные вопросы в связи с так называемыми принципами суперотбора. Несмотря на это, возможно ввести согласованную алгебраическую конструкцию таких правил, представленную в данной работе. Рассмотренные методы имеют определённую аналогию с конструкцией супер-пространств, но при этом обладают некоторыми отличиями от стандартных определений суперсимметрии иногда используемых в обобщенных моделях кубитов.
Ключевые слова: преобразование Жордана-Вигнера, кубиты с нетривиальными перестановочными соотношениями, квантовая информатика
Благодарности. Статья представлена на Конференции «Квантовая информатика - 2021», Москва.
QUANTUM AND MOLECULAR COMPUTING AND QUANTUM SIMULATIONS
1. INTRODUCTION
Analogues of fermionic creation and annihilation (ladder) operators were suggested by Richard Feynman for description of quantum computers already in the very first works [1; 2]. However, Jordan-Wigner transformation [3] is necessary to make such operators anticommutting for different qubits. Such approach was used later in so-called fermionic quantum computation [4].
Representation of fermionic ladder operators in such a way formally requires some consequent indexing (order) for description of Jordan-Wigner transformation. The order does not manifest itself directly in algebraic properties of ladder operators, but transformations of states formally depend on such indexes in rather nonlocal way.
States of physical bosons and fermions can be described in natural way by symmetric and antisymmetric tensors respectively, but fermionic quantum computation is rather relevant with more subtle exchange behavior of some quasi-particles.
Formally, qubits in state |0) corresponds to 'empty modes' and only qubits in state |1) treated as 'occupied modes' could be relevant to fermionic exchange principle for qubits marked by some indexes. Consistent mathematical model of such 'super-indexed' states is suggested in presented work.
More detailed description of such states is introduced in Sec. 2.1 together with formal definition of signed exchange rule and 'super-indexed' qubits denoted further as S-qubits. The different kinds of operators acting on S-qubits are constructed in Sec. 2.2.
Algebraic models of S-qubits are discussed in Sec. 3. The 'non-trivial' non-commutative part of such model is similar with exterior algebra recollected in Sec. 3.1. However, 'trivial' commutative elements could not be naturally presented in such a way and more complete model is suggested in Sec. 3.2. The Clifford algebras initially used in Sec. 2.2 for applications to gates and operators become the basic tools here. The S-qubits are introduced as minimal left ideals of the Cliford algebras. Finally, some comparison with possible alternative models of qubits related with super-spaces are outlined in Sec. 3.3.
2. SUPER-INDEXED QUBITS
2.1. STATES
Let us introduce special notation for qubits marked by some set of indexes I with basic states denoted as
1 -=fjL V, ...)' * * - -* v, ... e{0,!}. (1)
All indexes in the sequence a, b, ... must be different. The I can be associated with some nodes in multi-dimensional lattices, more general graphs or other configurations without natural ordering. Thus, the qubits in Eq. (1) may be rearranged in different ways.
An idea about basic states of qubits as 'occupation numbers' of anticommuting 'local fermionic modes' (LFM) [4] can be formalized by introduction of equivalence relation between elements Eq. (1) with different order of the indexes defined for any neighboring pair by signed exchange rule
I o\| _ I 6\| 0\ I a\I b\ _\b\I 0\ I a\I b\ _ I b\\ |o/|o/ " |o/|o/' |o/|l/ "\l/\o/' |l/|l/ " |l/|l/'
i.e.
I =:}fc); Vo * *e I' *-{M. (2)
Due to such rules terms |0) with 'attached' indexes can be exchanged ('commute') with any state |ф) = a|0) + p|l), but two |l) require change of the sign for such a swap ('anti-commute'). For standard notation and qubits without special indexes the exchange rule could be implemented by signed swap operator
r1 0 0 0 ^ 0 0 10 0 10 0 , 0 0 0 -1
S± =
(3)
States Eq. (1) with equivalence relation Eq. (2) define basis of some linear space S denoted here as 'super-indexed' qubits or S-qubits.
The equivalence relation Eq. (2) can be extended on arbitrary permutation n. Such operator is denoted further n ± and notation n is reserved for usual permutation. The construction of n ± does not depend on decomposition of n on adjacent transpositions, i.e., swaps of S-qubits considered above. Such consistency becomes more natural from algebraic models below in Sec. 3 and 'physical' interpretation with LFM.
It can be also proved for arbitrary state by direct check for the basis. Let us consider for a given basic state different sequences of transpositions produced the same permutation. It is necessary to show that the sign does not depend on the decomposition of the permutation into the sequence. Let us consider restriction (n1) of permutation on subset of indexes corresponding to S-qubits with unit value. For the only nontrivial case the restriction of swap on such subset corresponds to exchange of two units with change of sign. So, for any decomposition the basic vector may change the sign only if the permutation n1 is odd. Thus, n± is the same for any decomposition of n on transpositions defined by signed exchange rule Eq. (2).
The relation S± can be considered as a formalization of swap with two LFM denoted as '<' in [4]. It could be expressed as composition of usual exchange of qubits and 'swap defect' operator [4]
(4)
Perhaps, the term 'fermionic qubits' might be not very justified for the model considered here, because the property Eq. (2) would correspond to fermion for |l) ('occupied,' n =1) and boson for |0) ('empty,' n = 0).
Thus, S-qubits could be considered as quasi-particles ('fermisons') with combined statistics, because exchange rule instead of (±1) multiplier for bosons or fermions should use swap defect operator
1 о о о 1
о 1 о о
о о 1 о
о о о -1,
(5)
where a formal representation D = (-1)"°"b is used, where na and nb are analogues of occupation number operators defined for usual qubit as
ñ| v) = v|v>, ve{0, 1}, П--
0 0 0 1
(6)
The result of a swap Eq. (5) is defined in simple 'product' form Eq. (2) for the basis, but for arbitrary states the expressions are less trivial.
24
Computational nanotechnology
Vol. S. No. З. 2021
The 5-qubit also could be compared with an element of super vector space, but due to some subtleties outlined in Sec. 3.3 such approach should be discussed elsewhere.
The scalar product of 5-qubits states can be naturally defined for the equivalent sequences of indexes 5 in both ('bra' and 'ket') parts
(7)
The definition Eq. (7) does not depend on a sequence 5. Indeed, let us consider permutations of indexes n: S ^ S'. For the basic states a permutation may only introduce (±1) multiplier and the scalar product Eq. (7) does not change. It can be also checked directly for arbitrary states
Y n, n_
(8)
2.2. OPERATORS
Description of quantum gates with annihilation and creation ('ladder') operators was initially suggested by R. Feynman [1; 2]. However, despite of formal resemblance with Pauli exclusion principle for fermions
â+|0> = |1>; â|1> = |0>; ô|0> = â+ |1> = 0,
(8a)
they do not satisfy canonical anticommutation relation (CAR) for different qubits. Sometimes, usual qubits are compared with so-called 'hardcore' bosons, but it is not discussed here. It is considered instead, how ladder operators with CAR can be introduced for 5-qubits, see Eq. (14) below.
The creation operator afa can be defined for basic states taking into account exchange rule Eq. (2) of 5-qubits
It a R \ I a L r\
aaI ..., 0, .../ = |l......../,
1,
:)=o,
(9)
aj 0, ..) = 0, aa |l, ...) = | 0, ..).
(10)
A sign for application of operator aa to arbitrary position should be found using rearrangement of indexes together with signed exchange rule Eq. (2).
Let us denote ±a sign derived from Eq. (2) for expressions such as
I L a r\ I a L R \ U 1, ... = ±a |1......../,
(11)
where ±a = (-1)#L with #L = ^ e nt is number of units in sequence L of positions before 'a'. for simplicity L and R are omitted further. Finally, Eqs. (9, 10) can be rewritten
a01 ..., 0, ..) = ±a|..., 1, ...), aa1, ...) = 0; aa0, ...) = 0, aa|..., 1, ..) = ±a|..., 0, ...).
(12a)
(12b)
and a^ is Hermitian conjugation
ao|n0, ... , na_u 0, na+u ...} = (-l)S"" |n0, ... , na_u 1, na + u ...)
ao|no.....na -1, 1, na +1, ..) = 0. (13b)
The aa and a^ defined in such a way satisfy canonical anticommutation
^ ah} = {aa, a+J = 0; {aa, a+h} = 6„,1.
(14)
Let us now introduce Clifford algebra C€(2m) with 2m generators using operators Eq. (13)
ea = /{at + aa}; e' = aa - a*.
a 1 a a*' a a a
(15)
Operators of so-called Majorana fermionic modes coincides with Eq. (15) up to imaginary unit multiplier [4].
With earlier definitions of annihilation and creation operators it can be expressed for basis as
e01..., 0, ..) = ±0/|..., 1, ..), ea1, ..) = ±0/|..., 0, ...);
A A A A (16)
e'a U 0, ..) = +0 |..., 1, ..), e'a |..., 1, ..) = ±a |..., 0, ...),
where + = -(±a).
For consequent indexes j = 0, ... , m - 1 the Eq. (16) again correspond to Jordan-Wigner formalism with definition of complex Clifford algebra C€(2m, C) by tensor product of Pauli matrices [3; 5]
ej = il®—®® ®a" ® 1®---® 1 ;
j m - j -1
e'j = iâz ®-®âz ®ay ® 1 ®-® 1.
j m - j -1
(17)
where L and R correspond to arbitrary sequences before and after index 'a' respectively. The conjugated annihilation operator aa = (a^)+ in simpler case with index 'a' in the first position can be written
For consequent indexes a = 0, ... , m - 1 Eq. (12) is in agreement with usual Jordan-Wigner transformation [3; 4], i.e.,
aa\n0, ..., na_... 1, na+...) = (-lX?0"" |n0, ... , na_0, na +...) aa\n0.....n0-1, 0, n0+1, ...} = 0, (13a)
However, Eq. (17) directly introduces order of indexes unlike more abstract definitions of operators such as Eq. (12) and Eq. (16) respecting structure of 5-qubits without necessity of predefined order.
The linear combinations of all possible products with operators ea, e'a (or aa, a^) for given set 1 with m3 indexes generate Clifford algebra CC(2m3, C) with dimension 22mi. Thus, an arbitrary linear operator on space 5 can be represented in such a way, but unitarity should be also taken into account for construction of quantum gates on 5-qubits.
An alternative notation ea, = ea, a' e I' ~ 1 unifying two sets of generators from Eq. (15) into the single collection with doubled set of indexes 21 = 1 U 1' is also used further for brevity. Definition of Clifford algebra C€(2m3, C) can be written with such a set as
{ea, eb} = -20ab1, a, b e 21. (18)
and conjugation of elements as
e; = -ea, a e 21. (19)
The elements Eq. (15) generate C€(2m3, C) isomorphic with whole algebra of 2m x 2m complex matrices. The unitary gates may be expressed as exponents of Hermitian elements with pure imaginary multipliers discussed below.
Let us consider for some sequence L with l indexes from 21 a product of l generators
e, = e„ ... e
ai'
a,, ... , a, e 21.
(20)
QUANTUM AND MOLECULAR COMPUTING AND QUANTUM SIMULATIONS
Linear subspaces CC1'1 is introduced as a span of such products, eL e C6(l). The notation C€° and CI1 is reserved here for standard decomposition of CI as Z2-graded algebra with two subspaces corresponding to linear span of all possible products with even and odd l respectively [10]
C6(n) = CC>) ® CC». (21)
The square of element eL can be expressed as
(22)
e2 = (-:)* ; Çl = modl.
long to even subalgebra C€° that may be again treated as a Clifford algebra due to standard isomorphism Cl(n - 1) s CC°(n) [10]
CC(n - l) ^ CC°(n); eL —
eL, eL g CC°(n - l); eLe„, eL g CCl(n - l).
(3О)
Thus, a model with even number of generators in Hamil-tonians and universal subset of quantum gate with l = 2, 4 [4] is also described by Clifford algebra due to isomorphism C€°(2m) s CC(2m - 1).
All such elements are unitary with respect to conjugation operation [5]
eL = (-1)Sl eL; eLeL = 1. (23)
The construction of Hermitian basis is also straight forward
(i^eL) = (-i)* eL = (-i)* (-1)* eL = i"eL. (24)
An exponential representation of unitary operators is simply derived from Eq. (24) for arbitrary compositions of basic elements, e.g., for h, e C€(l)
u (t) = exp (-1' h't), i, = i ■ i= i+\ 16 R. (25) Due to property q,+4 = q, multipliers can be given in the table,
l mod 4 О l 2 3
il О l l О
-ll —i l l —i
(26)
Expression of unitary group U (2m) using families of quantum gates can be derived using approach with exponents due to correspondence between Lie algebras and Lie groups. The method initially was suggested for construction of universal set of quantum gates [6; 7; 8]. Clifford algebra C€(2m) with Lie bracket operation defined as a standard commutator
[o, b] = ob - bo
(27)
u
... = exp (h1e. + h2e' + h3e. ej), h1, h2, h3 e R. It can be also rewritten
u¡=qo+qle¡+q2e¡ + q3e¡e¡; qo + q2 + q2 + q3 = l, q<,, ql, q2, q3 g R.
(2S)
(29)
3. ALGEBRAIC MODELS OF S-QUBITS 3.1 EXTERIOR ALGEBRA
Let us consider a vector space V with basis x., j = 0, ... , m - 1.
The exterior (Grassmann) algebra is defined
Л(V)= e Лк (V),
к = 0
(3l)
where A°( V) are scalars, A1( V) = V are vectors, and Ak( V), k > 1 are antisymmetric k-forms (tensors) with basis
x, л л V ¡l < ••• < ¡k
(32)
where 'A' denotes antisymmetric (exterior) product x. A xk=-xk A x., x A x = 0, Vx e V.
The dimension of whole space A( V) is 2m and any basic state Eq. (1) of S-qubits could be mapped into A(V)
..... "J — Л X.
П; = l
(33)
can be used for representation of Lie algebra of special unitary group su(2m) and group SU(2m) can be expressed as exponents of elements from C€(2m).
In the exponential representation analogue of one-gates for S-qubits with l =1, 2 canbeexpressedas
Analogue of two-gates for S-qubits with indexes j, k e 1 can be declared by analogue exponents with linear combination of different products including e, e', ek, e'k with coefficients are either real (l = 1, 2) or pure imaginary (l = 3, 4).
Similar exponents with more general elements from linear subspaces C€(l) for l = 2 and l = 1, 2 (with arbitrary combinations of indexes from 21) generate 'non-universal' subgroups isomorphic to Spin (2m) and Spin (2m + 1) respectively [5; 8], but inclusion element with l = 3 is enough to generate unitary group SU(2m) [8].
Due to physical reasons for some models only terms with even number of generators must be used [4]. Formally, such terms be-
Such a method inserts into exterior product only x. with indexes j satisfying n. = 1. However, Eq. (33) is one-to-one map and arbitrary form Q e A(V) corresponds to some state |Q) up to appropriate normalization.
The creation and annihilation operators in such representation correspond to a known construction of Clifford algebra using space of linear transformations on A(V) [5] and may be expressed for basis Eq. (32)
a,: x, a---a x, ^ x, a x, a ■■■a x, ; (34a)
j Ji Jk j Ji Jk
k l
a;: xj a■■■ axjt ^ ^(-l) xj± a — a^ji)a---axj , (34b)
where 1 is unit of algebra A(V) and notation 1 A w = w A 1 = w, w e A(V) is supposed in Eq. (34). Such operators satisfy Eq. (14) and respect map Eq. (32) due to consistency of Eq. (34a) with Eq. (9) and Eq. (34b) with Eq. (12b).
The generators of complex Clifford algebra C€(2m, C) can be expressed with earlier defined pair of generators Eq. (15) for each index and for real case elements e' might be used to produce CC(m, R) [5].
The considered representation of S-qubits with exterior algebra A(V) despite of one-to-one correspondence Eq. (33) for complete basis may be not very convenient for work with 'reduced' expressions such as Eq. (2), because only qubits with state |1) map into different xa, but any sequence of qubits in state |0) formally corresponds to unit scalar 1 e A0(V). An approach with Clifford algebras discussed next helps to avoid such a problem.
26
Computational nanotechnology
Vol. S. No. 3. 2021
l=i
3.2. CLIFFORD ALGEBRAS AND SPINORS
For Clifford algebra CI = C€(2m, C) the space of spinors has dimension 2m and it can be represented as minimal left ideal [9]. The left ideal l c CI by definition has a property
ci e l: Vi e l; c e CC.
(35)
By definition, the (nonzero) minimal left ideal does not contain any other (nonzero) left ideal.
The notation with single set of indexes a e I and m pairs of generators ea and ea is again used below. Annihilations and creation operators corresponding Eq. (15) are also useful further
,e„ + iea 2i
a„ =-
2i
(36)
The minimal left ideal l is generated by all possible products with an appropriate element l0
l = R,: c e C1, l0 = f]l0
where
l0a
1 + ieae'a 2
(37)
(38)
e'l0 :
a0
-/e l0.
a0
Let us apply definition of l Eq. (37) to linear decomposition of c on terms with products of generators ea and ea. Any element of l in Eq. (37) may be rewritten as a linear combination of terms without e'a due to Eq. (39). Thus, l has dimension 2m with products at most m different generators ea on l0 as a basis. Let us also introduce notation
: ay0 --
(40)
then a basis of spinor space l can be rewritten in agreement with Eq. (1)
ll...)l0 ^ U v.
a, b, ...e 1, v, ... e {0, 1}, (41)
there all indexes a, b, ... must be different.
In representation Eq. (17) with consequent indexes a = 0, ... , m - 1 the elements 6a0 correspond to 2m x 2m diagonal matrices with units and zeros described by equation
■ ®1 ® l0 1, (42)
a m - a -1
where l0 = |0)(0|. Therefore, l0 corresponds to a 2m x 2m diagonal matrix with unit only in the very first position
l0 ^ «0 C
> «0 = |0, . . . , 0X0, ... , 0|.
(43)
In such a case a product cl0 in definition of ideal I Eq. (37) corresponds to a 2m x 2m matrix with only nonzero first column. It can be used for representation of a vector with 2m components.
For arbitrary sequences of indexes from a set I an analogue of Eq. (2) also holds
Il = (-1pgbga, a * b e I, v e {0, 1}. (44)
The inequality of indexes a * b is essential, because the super-commutativity Eq. (44) does not hold for a = b if | * v
e°e° = e°; i1ai1a = e°e° = 0; e°e° = e°; e°e° * e°ea0. (45)
Anyway, all indexes of 5-qubits Eq. (1) are different by definition and inequality in Eq. (45) does not affect considered representation Eq. (41).
However, some other expressions for operators or scalar product Eq. (7) may require more general combinations of indexes. It is discussed below.
For arbitrary element l e I the operators aa, a^ can be naturally defined via left multiplication
aa: l ^ aal; afa: l ^ a+al.
a a ' a a
With respect to map Eq. (41) it corresponds to Eq. (12). Let us check that.
Operators aa, afa commute with lbb and anticommute with
for a * b. For equivalent indexes quite natural expressions follow from definitions
a l0a : a0
0; aft = l0a; a+al0a = l- a+fil * 0.
(46)
0' a 0
Let us rewrite map Eq. (41) with shorter notation for basic states n
A A
IS = 1 "J. ' i ' ..) ^ ln=n j (41')
jE 1
It may be also expressed in an alternative form
are N commuting projectors (l^2 = l0a. Due to identity la = ie e'la for any index a it can be written
0 a a 0 1
(39)
ln =n(a y =fn a j
Aj
nI = 1
(47)
resembling Eq. (33) for Grassmann algebra. Due to Eq. (37) products of operators aa are mapped by Eq. (47) into elements of left ideal of Clifford algebra, cf Eq. (35).
With respect to map Eq. (41) actions of a^ are in agreement with Eq. (9) and aa satisfy an analogue of Eq. (10). Thus, operators a , a+ and their linear combinations e , e' are corresponding
a a a a
to Eq. (12) and Eq. (16) respectively.
The scalar product Eq. (7) also can be naturally expressed. Let us find conjugations of l0a and la
loa+ = aaa+a = la ia+ = aa, (48)
The equation
lja+lka = 5/0a; j, k = 0, 1 (49)
can be checked directly
la la = la la = la; la la = la la = 0
0 0 1 1 0' 0 1 1 0
together with appropriate expressions for products Eq. (41')
l+l = l0; l+l ' = 0. (50)
n n 0' n n v '
Let us also use notation lv for representation of arbitrary | i.e., linear combinations of basic states. Then scalar product Eq. (7) can be written using Eq. (50)
l l
o = M 0
l
(51)
"0- Wl,
where super-index I denotes set of indexes used in Eqs. (41), (41') and it can be dropped, because all indexes are naturally taken into account in such algebraic expressions with appropriate order.
Special notation can be used for 'scalar part' of an element
c e C€(2m; C); c = c1 -
Sc(c) = c.
(52)
Eq. (51) can be rewritten now to express the scalar product directly
Sc(i;i0) = Sc«^A O)l0) = <^AO)Sc(l0) = 2-m W O). (53)
Let us introduce an analogue of density operator. For pure state it can be defined
(54)
a
КВАНТОВЫЕ СТРУКТУРЫ И КВАНТОВОЕ МОДЕЛИРОВАНИЕ QUANTUM AND MOLECULAR COMPUTING AND QUANTUM SIMULATIONS
with natural property
= = ^ <Y|O)l0 = WO)«/0 = <Y|0)V (55)
Arbitrary operators can be expressed using linear combinations with pairs of basic states
in'Xnl ~ Pn,n = lX; Pn,nlO = «n-<n|®)«0 = ®A. (56)
3.3. SUPER VECTOR SPACES
A super vector space [11; 12] is Z2-graded vector space
V = V0 e V,; 0, leZ2 = —. 0 1 2 2Z
(57)
The complex super vector space is denoted C d°|d1, where d. is dimension of V. The elements v e V0, p(v) = 0 and w e V, p(w) = 1 are called even and odd respectively.
The Z2-graded (super) tensor product for such elements can be defined using sign rule
v & u = (-1)pM p(u) u & v (58)
Roughly speaking, one S-qubit could be compared with element of C1!1, but such approach encounters difficulties for more S-qu-bits. Indeed, Z2-graded tensor product Eq. (58) in definition Eq. (1) should use different copies of initial space. Such approach may be quite natural in definition of Z2-graded tensor product of algebras and can be used for construction of Clifford algebras [10; 13].
However, it is not quite clear, how to implement similar idea for construction of S-qubits from C1!1, because implementation of m different copies of S-qubit may require to use bigger spaces such as C m|m.
Let us consider basis of V = Cm|m: e° e V0, e\ e V1, k = 0, ... , m - 1. States of qubits ae°k + Pek belong to different 2D subspaces of V and their tensor product for k = 0, ... , m - 1 is the linear subspace with dimension 2m of the 'whole' tensor product V&m with dimension (2m)m. However, it may look as not very natural choice.
For more trivial cases Cd°'° and C0|d1 the tensor product of super vector spaces could be treated as symmetric and antisymmetric tensors respectively, but in such a case all vector spaces in product are usually considered as identical. Similar approach with identical copies of C d°|d1 is also quite common in supersymmetry. Thus, superspace is only briefly mentioned here for comparison with other models of S-qubits and the term super-indexed is used earlier to emphasize the difference with known supersymmetric model of qubits [14].
It should be mentioned also, that in the spinor model of S-qu-bits discussed in Sec. 3.2 the elements formally do not belong to superalgebra. Despite the super-commutation rule is valid for different super-indexes Eq. (44), it can be violated for the same one, Eq. (45).
4. CONCLUSION
Jordan-Wigner transformation maps some operators Eq. (8a) acting 'locally' on n qubits (or spin-1/2 systems) into n fermionic creation and annihilation (ladder) operators. The 'nonlocal' construction of such a map supposes introduction of some formal ordering on the set of qubits. Such ordering may be natural for some simple models such as 1D chain. All ladder operators in fermionic system are formally equivalent and unnatural order produces technical difficulties for more general models such as multidimensional lattices and more general graphs.
Antisymmetric algebra may be formally used for equal (unordered) description of ladder operators, but it does not answer a question about inequality of states. To address such a problem in this work was suggested model of 'super-indexed' S-qubits. Equivalence relation necessary for agreement with Jordan-Wigner transformation and anti-commutativity of ladder operators is signed exchange rule for S-qubits Eq. (2).
Algebraic model of S-qubits with such property was also discussed. The model uses Clifford algebras and spinors. Such approach is different with analogue constructions more common in supersymmetric models only briefly discussed in subsection above.
References
1. Feynman R.P. Simulating physics with computers. Int. J. Theor. Phys. 1982. No. 21. Pp. 467-488.
2. Feynman R.P. Quantum-mechanical computers. Found. Phys. 1986. No. 16. Pp. 507-531.
3. Jordan P., Wigner E. Uber das Paulische Aquivalenzverbot. Zeitschrift fur Physik. 1928. No. 47. Pp. 631-651.
4. Bravyi S., Kitaev A. Fermionic quantum computation. Ann. Phys. 2002. No. 298. Pp. 210-226. arXiv:quant-ph/0003137
5. Gilbert J.E., Murray M.A.M. Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge: Cambridge University Press, 1991.
6. DiVincenzo D.P. Two-bit gates are universal for quantum computation. Phys. Rev. A. 1995. No. 51. Pp. 1015-1022. arXiv:cond-mat/9407022
7. Deutsch D., Barenco A., Ekert A. Universality in quantum computation. Proc. R. Soc. Lond. A. 1995. No. 449. Pp. 669-677. arXiv:quant-ph/9505018
8. Vlasov A.Yu. Clifford algebras and universal sets of quantum gates. Phys. Rev. A. 2001. No. 63. P. 054302. arXiv:quant-ph/0010071
9. Lounesto P. Clifford Algebras and Spinors. Cambridge: Cambridge University Press, 2001.
10. PostnikovM.M. Lie Groups and Lie Algebras. Moscow: Mir Publishers, 1986.
11. Varadarajan V. Supersymmetry for mathematicians: An introduction. Providence: AMS, 2004.
12. Deligne P., Morgan J.W. Notes on supersymmetry (following Joseph Bernstein). Quantum Fields and Strings: A Course for Mathematicians, 1. P. Deligne et al. (eds.). Providence: AMS, 1999. Pp. 41-97.
13. Vlasov A.Yu. Quantum circuits and Spin (3n) groups. Quant. Inf. Comp. 2015. No. 15. Pp. 235-259. arXiv:1311.1666 [quant-ph]
14. Frydryszak A.M. Qubit, superqubit and squbit. J. Phys.: Conf. Ser. 2013. No. 411. P. 012015. arXiv:1210.0922 [math-ph]
Статья поступила в редакцию 07.06.2021, принята к публикации 14.07.2021 The article was received on 07.06.2021, accepted for publication 14.07.2021
ABOUT THE AUTHOR
Alexander Yu. Vlasov, P.V. Ramzaev Research Institute of Radiation Hygiene. Saint Petersburg, Russian Federation. ORCID: 0000-0002-6553-229X; E-mail: a_y_vlasov@ yahoo.com
СВЕДЕНИЯ ОБ АВТОРЕ
Власов Александр Юрьевич, научный сотрудник Федерального бюджетного учреждения науки «Санкт-Петербургский научно-исследовательский институт радиационной гигиены имени профессора П.В. Рам-заева» Федеральной службы по надзору в сфере защиты прав потребителей и благополучия человека. Санкт-Петербург, Российская Федерация. ORCID: 0000-0002-6553-229X; E-mail: a_y_vlasov@yahoo.com
2S
Computational nanotechnology
Vol. S. No. 3. 2021
