PhysComp96: umen Extended abstract Draft, 12 May 1996
Simulating quantum non-integrable systems with
umeno:
[email protected]
quantum computers
Ken Umeno3
The Institute of Physical and Chemical Research (RIKEN), FRP.
An impossibility theorem on approximately simulating quantum non-integrable Hamiltonian systems by quantum computers is presented here. This result is obtained by a quantum-mechanically interpretation of Ge-Marsden's impossibility result about simulating classically non-integrable Hamiltonian systems using integration schemes preserving symplectic (Lie-Poisson) property. Discussion is devoted to whether there is a rigorous connection between computability and integrability in the computational problem to simulate physical processes by physical computers.
there is a more general connection between computability and integrability at least in the problem of simulating Since the recent discovery of polynomial-time algo- physical behavior, which will be discussed in Section 4. rithms on prime factoring by quantum models of computation[12], a great attention is now directed to investigate the theoretical power of quantum computation 2 Explicit quantum algorithms in a more general setting. On the other hand, simulating quantum behavior such as quantum chaos using The hallmark of quantum computation is that each eleclassical computers is known to be a dicult computa- mentary dynamical process of the state vector of quantum tional problem. This computational diculty in simu- computers obeys the unitary transformation[1, 2, 3]: The lating quantum physics seems one of the principal mo- time-evolution operator has the form of tivations by Feynman to introduce an idea of quantum U (1t) = exp [0i1tH=~] ; (1) mechanical computers[5, 6]. On the contrary, this speculation raises the following computational problem: Can where H is a Hamiltonian operator with Hermitian propuniversal quantum Turing machines originally de ned by erty, 1t is the time duration of each computation process Deutsch[2] simulate arbitrary quantum mechanical behav- and an exponential operator exp [xA] of A is de ned as the sum of the convergent series ior itself? In this paper, I will give a somewhat negative answer 1 X to this question in the case of quantum non-integrable exp [xA] = (xAn!) ; x = 0i1t=~ (2) systems with an explicit time-independent Hamiltonian =0 operator: No explicit algorithm on the quantum model Let A and B be Hermitian operators as the generators of computation can simulate quantum non-integrable be- of two di erent elementary processes of quantum comhavior without violating, at least either the conservation putation. By di erent it means that A and B are nonlaw of energy expectation or the unitary property. Here, commutative operators, since the identity by explicit algorithm on the quantum model of computation it means that each elementary computational process exp [x(A + B)] = exp(xA) exp(xB) (3) must have an explicitly time-dependent Hamiltonian gen- is valid if and only if the commutator of A and B vanishes: erating each unitary transformation. Since the original quantum nature must have these two properties, namely, [A; B] = AB 0 BA = 0: (4) the conservation law of energy expectation and the unitary property of time evolution, whether it is quantum To track computational processes successively retaining integrable or quantum non-integrable, this means that unitary property, we have to evaluate the following timethere is an unavoidable limitation in simulating quantum evolution operator non-integrable behavior using quantum computers. This exp [x(A + B)] : (5) negative result has a classical counterpart given by Ge and Marsden[7]: No symplectic integrator can simulate non- Calculating (5) is relevant to various problems: integrable behavior in a class of autonomous Hamiltonian Feynman[4] discovered the formula called the Feynman systems without violating, at least either the energy con- path-integral method servation or the symplectic property. This suggests that (6) exp [x(A + B)] 1 + x(A + B) 1
Introduction
n
n
n
3 Supported
in part by the Basic Science Program at RIKEN
n
1
Let hus P consider ithe problem of approximately simulating exp x =1 A based on an explicit algorithm on quantum model of computation whose each elementary process is successively generated by explicitly time-dependent Hamiltonians Q ; 1 j m. Then, each s-th order approximation formula has a form:
in his study of quantum electro-dynamics based on the identity x( A + B ) : (7) exp[x(A + B)] = lim !1 1 + n However, the above approximation breaks unitary property in each elementary dynamical process 1 + ( + ) , because (1 + ( + ) )(1 + ( + ) )y = (8) (1 + 2( + )2)(1 0 ( + ) ) = ( + ) 1 0 2 6= 1: On the contrary, Trotter formula[16]
l
n
x A
j
2
exp 4x
B
n
x A
x A
B
n x A B
x
n
x2 n
based on the identity exp [x(A + B)] = lim !1
n
xB exp( xA ) exp( ) n n
xA
xB
n
n
xB
xA n
xA
n
3
i
=1
i
m Y
j
=1
exp [xQ ] + O(x +1 ): j
s
=1
A
j
=
m X
j
=1
Q
j
(13) (14)
Quantum
non-integrability
ef-
fect on quantum computation
Let us consider a time-independent Hamiltonian H ( ) which has no additional conserved operators in a certain class of the set of Hermitian operators G~ fG( )g, where denotes the canonical conjugate operators in the standard sense of quantum mechanics. Namely, [8; H ] = 0 =) 8 = F (H ); 8 2 G~ (15) for a function of a variable F . We call here this kind of quantum Hamiltonian systems H quantum nonintegrable. Since H is a time-independent Hamiltonian operator, the expectation value of H is preserved: q; p
n
q; p
n
q; p
n
q
=
must hold because the lowest order terms in x of L.H.S (9) and R.H.S. in Eq. (13) must be equal. In Section 3, we discuss the problem concerning this type of quantum algorithms having the expression (13) in the context of simulating quantum non-integrable systems.
x
p
5
j
j
(10) reserves the unitary property in each elementary process exp( ) exp( ), as it is easy to check it. The second order formula called leapfrog method has the form exp [x(3A+ B)] = 2exp( 2 )exp( ) exp( 2 )3 +O 2 ; (11) whose e ectiveness over the rst order method was con rmed in simulating the time-dependent Schrodinger equations[10]. Interestingly, there are many other higherorder formulas for exponential operators exp [x(A + B)] which were discovered independently both in the eld of quantum Monte Carlo methods[13, 15] to simulate quantum statistical mechanics where the exponential operators correspond to density matrices, or in the eld of symplectic integrators[11, 14, 15, 17, 24] to simulate classical Hamiltonian dynamical systems where the exponential operators corresponds to canonical transformations preserving the symplectic structure: X dp ^ dq : (12) !2 = d ^ d = n
A
l X
n
=1
3
The relation
n
) exp( xB ) +O exp[x(A + B)] = exp( xA n n
l X
j
B
n x A B
n A B
j
j
n
d d h H i = h jH j i = 0; dt dt
(16)
where < j is the state vector. Here, we obtain the following theorem: Theorem 1 If an explicit algorithm based on the quantum model of computation can simulate a quantum non-integrable system with a time-independent Hamiltonian H approximately, the conservation law of the expectation value of the Hamiltonian operator hH i breaks down.
i
;n
We can easily extend these decomposition formula of the exponential operators exp [hx(AP+ B)] toi more generalized exponential operators exp x =1 A of multi noncommutative operators A1; A2 ; 1 1 1 ; A . This means that there is no approximate algorithm to An application of these successive composition for- simulate quantum non-integrable systems without breakmulas of exponential operators to quantum computa- ing the unitary i h P property of each elementary process of tions of exp x =1 A is straightforward as follows: computation generated by an explicitly time-dependent l
j
j
l
l
j
j
2
Since we can choose 1T an arbitrary real number, the relation (22) means the following commutation relation: h i H; H~ (t) = [H; H (t)] = 0: (23)
Hamiltonian and the conservation law of the energy expectation. Since, as Deutsch suggests[3], the former condition that each elementary process of computation must have an explicitly time-dependent Hamiltonian is inferred to be critical for realizing universal quantum computing machines, we can guess that an actual simulation of quantum non-integrable systems is beyond the theoretical computation capacity of universal quantum Turing machines, i.e., it is a super quantum-Turing problem. Proof We prove this by reductio ad absurdum. With the use of the expression of quantum algorithms in Eq. (13), we have an expression of each s-th order algorithm of approximately simulating the quantum dynamics of H :
s
However, since H is assumed to be a quantum nonintegrable Hamiltonian, this commutation relation (23) means that the quantum algorithm H~ generates the exact quantum dynamics of H . This exactness (s ! 1) contradicts the assumption that the underlying quantum algorithm gives a approximately computing of H in the nite order s. Remark 1: This theorem does not depend on the order and types of approximation algorithms we choose. Remark 2: This proof is essentially same as the proof used in a paper [21] where the connection of the theorem Y (17) with the foundation of quantum mechanics is stressed by exp[xH ] = exp [xQ ] + O(x +1); illustrating a candidate of quantum non-integrable and =1 intermittent system shown in [22]. where x = 0i1t=~ and 1 s < 1. Each quantum algorithm Q (t; 1t) has the formula using corresponding time-dependent Hamiltonian H (t): 4 Concluding remarks R Main theme of the present paper is about the comQ (t; 1Pt) = T (exp R +1 H (sR)ds) = 1 1 n01 = 1 + =1(0 ~ ) 0 dt1 1 1 1 0 dt H (t1 ) 11 1 H (t ); putational limit of physical models of computation in simulating physical behavior. In Section 3, we give a theorem showing a limitation of quantum computation where T denotes the usual time ordering. The resulting quantum algorithm for the time duration concerning approximate algorithms of simulating quan1t has also an time-dependent (e ective) Hamiltonian H~ tum non-integrable systems. We can regard this theorem as a quantum analogue of Ge-Marsden's result telling us satisfying the relation a limitation of symplectic integrators to apply classical Y systems. However, this is not a trivial exexp [xQ ] = exp(xH~ ): (18) non-integrable tension of the classical version of the impossibility result =1 because of the notion itself of quantum integrability in the analysis. Although, our knowledge about exact After successively applying the Baker-Campbell- present criteria of classical non-integrability for explicitly given Hausdor formula: Hamiltonian systems have recently accumulated based on expX expY = expZ; (19) the singularity analysis[9, 18, 19, 20, 23, 25, 26], we lack our knowledge concerning exact criteria of quantum nonintegrability for explicitly given Hamiltonian operators: where We know that it is not a trivial thing to connect classical 1 1 non-integrability with quantum non-integrability[8]. Z = X + Y + [X; Y ] + ([X; [X; Y ]] + [Y; [Y; X ]]) 2 12 Thus, the present analysis has a problem to be solved 1 whether we can give an explicit quantum non-integrable + 24 [X; [Y; [Y; X ]]] + 1 1 1 (20) Hamiltonian system or not. We believe that there are to the system (18), we can compute the corresponding many candidates of quantum non-integrable systems like a quantum version of three-body problems such as time-dependent Hamiltonian H~ in a form: a helium showing irregular quantum spectra. Besides ~H ( ; t) = H + H (t) = H + O(x ); (21) the problem between quantum non-integrability and classical non-integrability, Theorem 1 combined with Ge~ where H (t) is a time-dependent correction term of H . Marsden's result seems to have an interesting implication We assume that the energy expectation < H~ > in the to the theory of computation and physics: Computability would be deeply connected with integrability. The quantum simulation is also preserved: conclusion of the present paper is that the assertion hH i = hH~ i = Const. for 0 t 1T: (22) holds, at least, when we consider approximate algorithms m
s
j
j
j
j
t
j
i
t
n
t
j
t
t
n
n
j
j
n
m
j
j
q; p
s
s
s
3
of simulating physical mechanical behavior using physical models of computation, whether it is quantum or classical.
[18] [19]
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