The digital quantum simulation of the dynamics of the SYK models involves fermionic operators, either Majorana or complex. In current technologies, such a quantum algo- rithm requires encoding fermionic operators into spin-1/2 operators. This is achieved via the JW transformation, as presented in section 2.1, from spinless complex fermion
1See a detailed derivation in appendix D.1. 2See appendix D.1.
operators to spin-1/2 operators, c†
i = (r i≠1
j=1‡zj)‡i+. We consider the relations be-
tween n complex fermions and N = 2n Majorana fermions, cj = ‰2j≠1 + i‰2j/2
and c†
j = ‰2j≠1≠ i‰2j/2. Thus, the corresponding Majorana fermionic operators are
codified as ‰2n≠1 = (rn≠1
j=1 ‡jz)‡nx and ‰2n = (r n≠1
j=1 ‡zj)‡yn, with {‰i, ‰j} = 2”ij.
Majorana interaction terms appear in terms of spin degrees of freedom as
‰i‰j‰k‰l = ‡˜i–i Q a˜i≠1Ÿ m=˜j ‡zm R b ‡–j ˜j ‡–˜kk Q a˜k≠1Ÿ m=˜l ‡zm R b ‡–l ˜l , (3.4) and ‰i‰j = ‡˜i–i Q a ˜i≠1Ÿ m=˜j ‡mz R b ‡–j ˜j , (3.5)
where i > j > k > l. Here, the tilded variables are
˜x =7x+ 12 8= maxÓmœ Z | m Æ x+ 12 Ô, (3.6)
and the –nlabels correspond to x if n is odd and y if even. Among the resulting spin
interaction terms, the most general and complex form corresponds to that shown in Eq. (3.4). In some specific cases of combination of indices the expression is simplified1.
Let us now consider the model with complex spinless fermions. The interaction terms can be mapped as above to spin interactions via the JW transformation. Thus, the interaction terms of type (i) of this model are expressed as
c†ic†jckc¸= Ÿ Q a ’Ÿ2≠1 ›=’1+1 ‡›z R b Q a ’Ÿ4≠1 ›=’3+1 ‡z› R b ‡i+‡j+‡k≠‡≠¸ , (3.7)
where {’1, ’2, ’3, ’4} = {i, j, k, ¸} as sets, ’1 < ’2 < ’3 < ’4, and Ÿ = sign(i ≠
j)sign(¸ ≠ k). For the sake of simplicity in the quantum simulation, we have only
taken into account the terms such that i > j and k > l, wherefore Ÿ = ≠1.
The interaction terms of type (ii), (iii) and (iv) can also be mapped to spin interactions as (ii) c† injck = ≠12 Q a ’Ÿ2≠1 ›=’1+1 ‡z› R b(‡z j + 1)‡i+‡k≠, (3.8) (iii) ninj = 1 4!1 + ‡zi + ‡jz+ ‡iz‡jz " , (3.9) (iv) ni= 12(1 + ‡zi), (3.10)
where {’1, ’2} = {i, k}, again as sets, and ’1 < ’2. It is still possible to reduce the
number of interaction terms by considering the properties of coefficients Jij;k¸1.
These spin Hamiltonians are a sum H = qm
i Hi, with Hi a many-body spin
interaction. A purely analog quantum simulation for the exact evolution is a difficult problem in any quantum platform. On the other hand, each spin interaction term can be handled individually in digital quantum simulations [22]. We recall the theory presented in section 2.1, and decompose the evolution operator in a Trotter–Suzuki product formula with s number of steps,
e≠iHt= Q a m Ÿ j=1 e≠iHjt/s R b s +ÿ i<j [Hi, Hj]t2 2s + O(J3t3/s2). (3.11) This expression approximates the dynamics for time t to an accuracy ‘ of the order of
J2t2/s. We note that for each non-zero commutator, [Hi, Hj] ”= 0, there is a decrease
in accuracy. In the worst case scenario, where all the commutators differ from zero, there will be a reduction of accuracy given by the factor !m
2", with m the number of
interaction terms Hi in the Hamiltonian.
The complexity of the algorithm, i.e., the number of gates required for the dy- namics simulation, grows polynomially with the number of fermions N. The coarsest evaluation suggests that achieving an accuracy ‘ over an evolution time t will require a number of gates m ◊!m
2"◊ J2t2/‘. In the case at hand, there will be m ≥ O(N4)
spin interactions. If each interaction is given by O(1) gates, the number of gates for accuracy ‘ over a time t will be O(N12). In fact, the number of non-zero commutators
is of order O(N6), rather than m2 ≥ O(N8), thus bringing the number of gates down
to O(N10). As usual, higher order Trotter–Suzuki decompositions, as the symmetric
expansion considered later in section 5.1, will improve the accuracy of our simulation with the cost of increasing the number of gates per step. Digitizing the evolution and its translation to a quantum algorithm enable the application of error correction techniques if the gates reach the fault-tolerant threshold [116, 117]. In principle, in such an error-corrected simulation the number of gates is unlimited, and thus our protocol gets to be scalable.
3.2.1 Protocol for correlation measurements
In order to probe the non-equilibrium behavior of the SYK, and more specifically, the dynamics of scrambling [109] in terms of OTO functions, we consider an effi- cient protocol for determining n-time correlation functions [118]. Here, an ancillary qubit QA encodes a correlation function by means of controlled operations. This ap-
proach is particularly effective for analog quantum simulation of the evolution, but it is also applicable to digitally synthesized quantum evolutions. This leads to the desired measurement of the four-time correlation function ÈW†
S(t)VS†(0)WS(t)VS(0)Í
as (ȇxÍ + iȇyÍ)A over the ancilla2. A similar approach in this context has been
1See appendix E.1 for a detailed derivation. 2See appendix E.2.
recently proposed [119]. Note that in order to evolve the system one requires time inversion, from t > 0 back to 0.
3.2.2 Protocols for time inversion
We need a time inversion operation for reversing the evolution of the system. Since the models are described by time independent Hamiltonians, reversing the sign of all the couplings gives us U(≠t), where U(t) denotes the time-evolution operator of the system. Alternatively, time inversion can also be implemented without explicitly engineering the algorithm for U(≠t)1. We consider an additional control qubit Q
C,
whose state decides the direction of the evolution in the system S as
UCS(t)| Í = –|eÍCU(t)|ÂÍS+ —|gÍCU(≠t)|ÂÍS, (3.12)
for an initial state | Í = (–|eÍC + —|gÍC)|ÂÍS. For an analogous construction, see
Ref. [121].
3.2.3 Protocol for state initialization
Scrambling depends on the Hamiltonian structure for typical initial states. It is possible to prepare thermal states on a quantum computer following existing methods in the literature [122, 123]. Moreover, it is also possible to analyze scrambling for explicitly known initial states, where a state with a certain number of excitations in localized fermionic sites can be constructed with single-qubit rotations2.