Description:FORM-2
THE PATENTS ACT, 1970
(39 of 1970)
&
THE PATENTS RULES, 2003

COMPLETE SPECIFICATION
(SEE SECTION 10, RULE 13)

A PSEUDORANDOM STATE GENERATION FOR SECURE CRYPTOGRAPHIC SYSTEM AND METHOD THEREOF

ARQANUM TECHNOLOGIES PRIVATE LIMITED,
AN INDIAN START-UP COMPANY HAVING OFFICIAL ADDRESS AT RATHOD HOSPITAL, ARAS LAYOUT, JAMBHRUN ROAD, BULDHANA, MAHARASHTRA, 443001, INDIA

THE FOLLOWING SPECIFICATION PARTICULARLY DESCRIBES THE INVENTION AND THE MANNER IN WHICH IT IS TO BE PERFORMED.

TECHNICAL FIELD
The present disclosure described herein, in general, relates to a system for processing quantum computing techniques and in particular for generating permutations using quantum computing techniques. Further, the present disclosure relates to a system for deploying quantum computing techniques on processing units.
BACKGROUND
Pseudorandomness plays a very important role in classical cryptography. Pseudorandomness is generally defined as a distribution which would be efficiently preparable by a given technique when provided with a key and such that it would be indistinguishable from a truly random distribution for any technique running in poly time. Quantum computers can efficiently prepare distribution of truly random numbers. It is also possible to create distribution of truly random quantum states called Haar random states. Quantum t-designs can be used to replicate the Haar random distribution over a polynomial of degree t or less. This led to the development of the unitary t-matrices which are considered ‘pseudorandom’ but are actually t-wise independent random variables. Recently, the notion of computational pseudorandomness was extended to the domain of quantum state ensembles, in the form of pseudorandom quantum states (PRS). These are computationally efficiently preparable states which are indistinguishable from the Haar random state by any technique running over polynomial time even after being provided with polynomially many copies of the states. The notion of PRS can be extended to pseudorandom unitaries (PSU) and even pseudorandom matrices (PRM). The idea of PRS is based on notion of one-way functions which are functions where a system can efficiently determine the output given the technique and the input, however the reverse will be a computationally hard problem. This property makes these states ideal for cryptographic tasks. One possible method of constructing pseudorandom quantum states involves the use of pseudorandom functions (PRF) and computing the phases of uniform superposition states. Alternative methods of constructing pseudorandom states can also be developed as long as one has access to the one-way functions.
The PRS provides a wide range of applications in quantum finance, including Quantum Money. Quantum Money requires that the quantum state (representing a unique note) cannot be efficiently cloned. The PRS obey the cryptography no-cloning theorem, which makes them viable for the Quantum Money scheme. There have been recent works discussing the robustness of PRS based cryptography schemes against adversaries. It was shown that the PSU can be constructed using quantum physical unclonable functions (PUFs), which makes them computationally hard for any adversary to clone.
The implementation of the PRS thus boils down to the choice of the quantum secure PRF used in constructing it. A simple way of implementing a quantum secure PRF is to use Hamiltonian dynamics which exhibit chaotic behavior. One such implementation of the PRS is done via the Sachdev-Ye-Kitaev (SYK) model used in AdS/CFT. The SYK Hamiltonian is highly scrambling in the sense that any perturbations would be dissipated throughout the system in a relatively short amount of time. This makes the time evolution chaotic. However, on its own the SYK time evolution does not give a PRS. While simulating quantum dynamics can take exponential time for a classical algorithm, a quantum computer can efficiently solve it in poly time. So, while this method satisfies the first condition of pseudo-randomness, it fails to satisfy the second on its own. To account for the technical problem, there is a need of a system and method that can choose to interject the time evolution with occasional perturbations.
Therefore, there is a need for a system and method which processes the quantum computing techniques for generating permutations to overcome the above limitations.
SUMMARY
The present disclosure relates to a system for deploying quantum computing techniques on processing units and particularly on method for generating pseudorandom quantum states (PRS) using the SYK Hamiltonian and the AdS/CFT correspondence. The method includes encoding a binary input message onto a quantum register, and applying N/2 pairs of 2-qubit SYK Unitaries (U_SYK) in parallel on the quantum register. Further, the method includes applying shock operators alternately on the qubits based on a secret key to generate permutations and performing measurements in the computational basis to estimate state amplitudes. The method further includes selecting an outcome with maximum probability as the output of a quantum circuit, generating an input-output map for a fixed key, and generating permutations for cryptographic applications using the input-output map.
In another embodiment, a system for generating pseudorandom quantum states is disclosed. The system includes a quantum circuit processor with a quantum register for encoding binary input messages, applying 2-qubit SYK Unitaries, and utilizing shock operators based on a secret key. The system further includes a classical processing/storage device with a classical register for storing measurement results obtained from the quantum circuit, and a permutation generation module configured to create input-output maps by running the quantum circuit multiple times, wherein the system configured for generating permutations for cryptographic applications, including S-Boxes for encryption/decryption circuits.
BRIEF DESCRIPTION OF THE DRAWINGS
The foregoing detailed description of embodiments is better understood when read in conjunction with the appended drawings. For the purpose of illustrating the disclosure, there is shown in the present document example constructions of the disclosure; however, the disclosure is not limited to the specific methods and apparatus disclosed in the document and the drawings.
Figure 1 illustrates a network implementation of a system for processing computing techniques, in accordance with an embodiment of the present disclosure.
Figure 2 illustrates a system diagram depicting the system of Figure 1 having different modules, in accordance with an embodiment of the present disclosure.
Figure 3 illustrates a schematic diagram depicting SYK based shock state quantum circuit architecture for pseudorandom quantum states generation, in accordance with an embodiment of the present disclosure.
Figure 4 illustrates a schematic diagram depicting a system for processing quantum computing techniques of Figure 1, in accordance with an embodiment of the present disclosure.
Figures 5a-5d illustrate schematic diagrams depicting forward quantum circuits for N, L = 2, 4, 6, 8 bits, in accordance with an exemplary embodiment of the present disclosure.
Figures 6a-6d illustrate schematic diagrams depicting inverse quantum circuits for N, L = 2, 4, 6, 8 bits, in accordance with an exemplary embodiment of the present disclosure.
Figure 7 illustrates a schematic diagram depicting a quantum circuit for hardware implementation, in accordance with an exemplary embodiment of the present disclosure.
Figure 8 illustrates a schematic diagram depicting a decomposed quantum circuit corresponding to a unitary operator, in accordance with an exemplary embodiment of the present disclosure.
Figure 9 illustrates a block diagram depicting a SYK shock circuit PRS generator-based quantum Merkle tree, in accordance with an embodiment of the present disclosure.
DETAILED DESCRIPTION
Various modifications to the embodiment will be readily apparent to those skilled in the art and the generic principles herein may be applied to other embodiments. For example, although the present disclosure will be described in the context of a system for processing quantum computing techniques and method thereof, it will readily recognize that the method and the system can be utilized in any situation where there is need to provide transmission of data from one node to other nodes. Thus, the present disclosure is not intended to be limited to the embodiments illustrated but is to be accorded the widest scope consistent with the principles and features described herein.
In one of the embodiments, the present disclosure discloses a system for processing computing techniques. The present disclosure discloses the generation of permutations using the AdS/CFT correspondence. The present disclosure includes a system and a method for generating the PRS which can be implemented on any gate-based quantum hardware. An input-output map has been generated based on the different keys. This input-output map gives the permutation for some particular unitary, which are generated from the SYK Hamiltonian. The present disclosure also provides a groundwork for potential applications of the AdS/CFT for the cryptographic protocols.
In an embodiment, the present disclosure discloses generation of pseudorandom quantum states (PRS) using the SYK Hamiltonian. Upon showing that the generated states are indeed pseudorandom, the system and method of the present disclosure perform a measurement in the computational basis to get a distribution to estimate the amplitudes of the state. Thereafter, the system picks an outcome with the maximum probability and chooses the same as the output of a quantum circuit. Noting down all such outputs corresponding to each input for a fixed key will give the full input-output map for that key. In this, the key determines the sequence of shock operators to be applied. The distribution is determined by running the circuit a fixed number of times for every input state. The larger the number of iterations, the more accurate estimation of the amplitude will be provided by the system.
In another embodiment, this input-output map for some particular values of the parameters of the time evolution unitary provides a permutation. This will be valid for cases where the system observes that the same input will give the same outcome with the maximum probability for the fixed key, i.e., for a fixed key a given input will always map to the same output, and that no two distinct inputs will map to the same output. After showing the generation of permutations using the SYK time evolution with the shock operators, the system will show that one can use these permutations for cryptography as potential S-boxes. The present disclosure thus also lays the groundwork for potential applications of the AdS/CFT conjecture for cryptographic purposes.
In another embodiment, a system for deploying quantum computing techniques on processing units and particularly on method for generating pseudorandom quantum states (PRS) using the SYK Hamiltonian and the AdS/CFT correspondence. The method includes encoding a binary input message onto a quantum register, and applying N/2 pairs of 2-qubit SYK Unitaries (U_SYK) in parallel on the quantum register. Further, the method includes applying shock operators alternately on the qubits based on a secret key to generate permutations and performing measurements in the computational basis to estimate state amplitudes. The method further includes selecting an outcome with maximum probability as the output of a quantum circuit, generating an input-output map for a fixed key, and generating permutations for cryptographic applications using the input-output map.
In another embodiment, a system for generating pseudorandom quantum states is disclosed. The system includes a quantum circuit processor with a quantum register for encoding binary input messages, applying 2-qubit SYK Unitaries, and utilizing shock operators based on a secret key. The system further includes a classical processing/storage device with a classical register for storing measurement results obtained from the quantum circuit, and a permutation generation module configured to create input-output maps by running the quantum circuit multiple times, wherein the system configured for generating permutations for cryptographic applications, including S-Boxes for encryption/decryption circuits.
Figure 1 illustrates a network implementation (100) of a system (102) for processing computing techniques, in accordance with an embodiment of the present disclosure.
Although the present disclosure is explained considering that a system for processing quantum techniques (hereinafter referred to as “system”) (102) is implemented on a variety of computing systems, such as a laptop computer, a desktop computer, a notebook, a workstation, a mainframe computer, a server, a network server, a cloud-based computing environment and the like. It will be understood that the system (102) may be accessed by multiple users through one or more user devices (104-1), (104-2), (104-3), (104-N).
In one implementation, the network (106) may be a wireless network, a wired network or a combination thereof. The network (106) can be implemented as one of the different types of networks, such as intranet, local area network (LAN), wide area network (WAN), the internet, and the like.
Figure 2 illustrates a system diagram (200) depicting the system (102) of Figure 1 having different modules, in accordance with an embodiment of the present disclosure.
In an embodiment, the system (102) may include at least one processor (202), an input/output (I/O) interface (204), and a memory (206). The at least one processor (202) may be implemented as one or more microprocessors, microcomputers, microcontrollers, digital signal processors, central processing units, state machines, logic circuitries, and/or any devices that manipulate signals based on operational instructions. Among other capabilities, the at least one processor (202) is configured to fetch and execute computer-readable instructions stored in the memory (206). Further, the I/O interface (204) may include a variety of software and hardware interfaces, for example, a web interface, a graphical user interface, and the like. The I/O interface (204) may allow the system (102) to interact with the user directly or through the user devices 104.
The memory (206) may include any computer-readable medium or computer program product known in the art including, for example, volatile memory, such as static random-access memory (SRAM) and dynamic random-access memory (DRAM), and/or non- volatile memory, such as read only memory (ROM), erasable programmable ROM, flash memories, hard disks, optical disks, and magnetic tapes. The memory (206) may include or be communicatively coupled to modules (208) and data (210).
The modules (208) include routines, programs, objects, components, data structures, etc., which perform particular tasks or implement particular abstract data types. In one implementation, the modules (208) may include a permutation generation module (212), simulation module (214) and other modules. The other modules may include programs or coded instructions that supplement applications and functions of the system (102). The modules (208) described herein may be implemented as software modules that may be executed in the cloud-based computing environment of the system (102).
The data (210), amongst other things, serves as a repository for storing data processed, received, and generated by one or more of the modules (208). The data (210) may also include system data (216) and other data (218). The other data (218) may include data generated as a result of the execution of other modules, and system data (216) may include data generated as a result of the execution of a permutation generation module (212), simulation module (214) in the other modules.
Figure 3 illustrates a schematic diagram depicting SYK based shock state quantum circuit architecture (300) for pseudorandom quantum states generation, in accordance with an embodiment of the present disclosure.
In Figure 3, an SYK based shock state quantum circuit architecture for PRS generation is shown. The SYK based quantum circuit architecture (300) includes a quantum circuit processor (302), a classical processing/storage device (304), a simulation module (214), and quantum hardware (306).
The quantum circuit processor (302) includes Quantum register 'Q' with Initial state, where all the N qubits Q(0), Q(1), ... , Q(N 1) are in|0> configuration and indexing from 0 to N 1; message/input 'm' encoded onto the N qubits according to the rule and indexing from 0 to N - 1. If m(i) = 1, apply X, else apply I; N/2 pairs of 2 qubit SYK Unitaries (U_SYK) applied in parallel on every jth iteration over the quantum register Q, where the starting point is j = 0 and ending point is j = N1 and each U_SYK = exp (iH_SYKt) where 'exp' is exponential function, 'i' is imaginary, 'H_SYK' is 2 qubit SYK Hamiltonian and 't' is time evolution value; and O1, O2, ..... , O(N-1), O(N) represent the N shock operator sequence, applied alternatively, following the rule that for jth SYK Unitary application, if K(i) = 1, apply X on ith qubit and if K(i) = 0, apply Z gate on ith qubit and indexing from 1 to N. The classical processing/storage device (304) includes classical register 'C' with N classical bits to store the measurement results and indexing from 0 to N1; and N Measurement operators 'M' in the Pauli Z basis applied on qubits Q(0), Q(1),..., Q(N-1) and stored in the classical register bits C(0), C(1), ... , C(N 1), respectively. In one embodiment, by using the stored results, the system (102) provides probability distribution to further provide output quantum basis state with maximum probability, and PRS state for storing maximum probability state. In another embodiment, the simulation module (214) may include an Aer Qiskit Simulator, which is having a quantum circuit simulated on a classical processing/storage device (304).
Further, the system (102) may be also configured to implement quantum hardware (306), which can be an AWS Rigetti Quantum Hardware of the quantum circuit, simulated on the quantum circuit processor (302). In an embodiment, the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, popularly known as ”Holography”, is a hypothetical connection between two different physical theories. Anti-de Sitter spaces (AdS), which plays an important role in theories of quantum gravity expressed in terms of a string theory or M-theory, are present on one side. Conformal field theories (CFT), which are a particular type of quantum field theories, describing various important physical phenomena, are on the other side of the correspondence. The duality is a significant improvement in a comprehension of quantum gravity and string theory. This is due to the fact that it is the most effective use of the holographic principle and that it offers a non-perturbative formulation of string theory with certain boundary conditions.
The Sachdev-Ye-Kitaev (SYK) model is suited to describe certain models of two-dimensional gravity which is exactly solvable albeit of the fact that it is a model of understanding certain aspects of quantum gravity.
SYK Hamiltonian: Let N be an integer and q an even integer such that 2 = q = N. Consider a set of Majorana fermions ?1,...,?N satisfying,
?†i = ?i,
{?i , ?j} = 2dij.
The SYK Hamiltonian is defined by,
………….(1)
where Ji1...iq be random variables whose expectations satisfy

In order to decrease the complexity, consider N = 8 and q = 4 SYK system. The Hamiltonian for (0+1)d model with m = 4 is given by:
, ………(2)
where ?i’s represent Majorana fermion operators. The coefficients Jijkl is anti-symmetric gaussian random tensors with Jijkl¯ = 0 and and J2 have the dimension of energy. The low temperature state in pure a SYK state, i.e., N >> J/T >> 0 represents maximally chaotic non-fermi liquid, which can be used to generate pseudo-randomness.
As the system (102) uses N = 8 model with q = 4, the system (102) includes 8C4 = 70 combinations of ?i ?j ?k ?l in the Hamiltonian. Using Jordan-Wigner transformation,
……… (3)
to rewrite the Hamiltonian as,
……… (4)
with a = {0,x,y,z} and s0 = I.
In order to implement this in into a computing platform, the system (102) has to discretize the time variable using a Suzuki-Trotter formula. The decomposed time evolution operator can be written as
……… (5)
where .

……… (6)
If the commutation relation in a decomposed time evolution operator vanishes, i.e., [Hs,Hs'], the accuracy in first-order approximation is boosted. The system (102) receives the fidelity to be 0.99 for ln(t) = 2 and ln(n) = 1.55. The fidelity always remains above 0.99 for ln(n) = 5.
Generation of permutations using AdS/CFT shock operator based on secret key:
Pseudo-random Quantum State Generation: PRS is defined as states that satisfy the following two conditions:
Efficient Generation: The state must be efficiently preparable for a polynomial time quantum technique G when given the key k. The algorithm must thus take the key k as the input and give the pseudorandom quantum state as output, G(k) = |?k?.
Pseudorandomness: For any efficient technique, the state must be indistinguishable from a Haar random state (truly random state) even when provided with polynomial many copies of the state.
Any ensemble of states that satisfies these conditions can be considered to be pseudorandom. In an embodiment, the system (102) may be configured to create such an ensemble using the evolution of the SYK Hamiltonian interjected with permutations in the form of “shock operators”, as shown in Figure 3. In Figure 3, The SYK Hamiltonian is special in the sense that it is considered to be ‘scrambling’. The SYK Hamiltonian can diffuse the perturbations through the systems in a very short time. This property however does not generate pseudorandomness, since ideally the time evolution can be reversed, and one can obtain the original state back efficiently. This is countered by introducing random perturbations throughout the evolution, which would get diffused in a short time resulting in a different state than what would have been using just the SYK Hamiltonian.
……………..(7)
Here, the system (102) uses the thermofield double state (TFD) as the initial state. TFD is a maximally entangled state of n qubits, where n is the entropy of the state (a black hole). However, this construction can be used to generate pseudorandom states using any state as an input.
Computationally this can be implemented by applying the unitary corresponding to the time evolution of the SYK Hamiltonian, followed by a shock operator and then repeated with each time the shock operator chosen at random. In this, the system (102) chooses a shock operator to be a Pauli operator acting on one of the qubits. Each shock operator corresponds to a branch in the evolution, thus after l shock, the final state can potentially be in one of the 2l states.
Pseudorandomness of the model:
The system (102) may be configured to prove the pseudorandomness of the SYK with the shock operator model. The argument given is that the problem of getting the initial state back from the final state is exponentially hard with the number of steps, since the output can be any one of the 2k possible outputs after applying k shock operators and guessing the initial state thus becomes an exponential search problem.
Implementation:
In an embodiment, the system (102) generates pseudorandom states using the time evolution of the SYK Hamiltonian with the shock operators, the system (102) shows the technique used for the same. The system (102) may be configured to implement the following operation
………..(8)
This is nothing but a sequence of unitary operations (gates) and can thus be implemented on a quantum computer. In one embodiment, the system (102) implements the SYK model using the Suzuki Trotter method, which is approximation of the actual time evolution operator.
, …………(9)
The system (102) uses the Hamiltonian at least l times, the errors in the approximation would propagate. However, the system (102) can counter this by taking smaller time steps instead. The errors depend on t3 and thus the errors propagated through multiple instances of the approximation would to some extent be countered by the relatively smaller size of the errors due to smaller time intervals. The design of the quantum circuit was inspired by Block ciphers, which take two inputs, one would be the message to be encoded and the other would be the key. In this technique as well, the system (102) will have two inputs, one would be the initial state and one would be the sequence of shock operators, which would act as the key. In block ciphers, the size of both the inputs is usually the same, thus in this circuits, the system (102) will limit the length of a key to the number of qubits.
Consider that the system (102) has n qubits in a circuit, i.e., a n qubit input. The key will be denoted by a n bit string. The system (102) can assign the Pauli operations to perform if the bit is 0 or 1. Here since the key is going to remain the same for multiple input messages, the system (102) can just manually prepare the circuit according to the key. Let the key be k = knkn-1kn-2 ...k2k1 such that ki ? {0,1}. Here, ki is a bit which implements a Pauli operation on the ith qubit. Thus, in one iteration of the circuit, each qubit will be subjected to a shock operator only once. The system (102) limits an action of the shock operator to once per qubit because if the same Pauli operator acts on the same qubit twice, the system (102) will get back the unperturbed qubit state. Limiting the action of the Pauli operation to once per qubit ensures that the system (102) gets the maximum perturbation while also minimizing the circuit depth.
Figure 4 illustrates a schematic diagram (400) depicting a system (102) for processing quantum computing techniques of Figure 1, in accordance with an embodiment of the present disclosure.
In Figure 4, a permutation generation module (212) is described which may be configured to cooperate with the SYK based shock state quantum circuit architecture (300) by using a hybrid classical-quantum circuit incorporated into the permutation generations scheme. The permutation generation module (212) may be configured to generate permutation for hybrid classical quantum circuit. The hybrid classical quantum circuit includes a hybrid quantum processing unit (QPU)/classical processing unit/system (402). The hybrid quantum processing unit (402) transmits a N-bit binary input message and L-bit binary secret message (404), to a permutation generator circuit (406). The permutation generator circuit (406) provides N-bit input-output L-bit secret key permutations, as shown at a block (410). These key permutations (410) are further used for applications (410), such as cryptographic primitives (412), S-Boxes (414), encryption/decryption circuit (416), quantum block ciphers (418), public-private key infrastructures (420), PRFS (pseudorandom function like states) (422), quantum hash functions (424), and the like.
In an exemplary embodiment, the system (102) may be configured to generate permutations based on secret key Parameters, where,
Unitary Operator U(t), without Suzuki Trotter Decomposition, is defined as:

where, the Hamiltonian with M terms are given by and each term Hs of the Hamiltonian belongs to a N dimensional Hilber space, and t denotes time evolution, Hs ? h1 ?h2 ?...?hN and each hi ? {|0?,|1?}.
S refers to number of quantum shots.
Key space: Set K = {k|k ? {0,1}L}
Message space: Set M = {m|m ? {0,1}n}
Assumptions:
Key size is equal to message size n.
Inputs:
Secret key k and message m.
Algorithm:
Initialize quantum register Q with message m. For i ? [n]:
if m[i] = 1 apply X gate on ith qubit else apply I gate.
Choose secret key k randomly from key space K.
For j ? [n]:
– For i ? [1,n,2] where 2 denotes the steps for i:
* Apply unitary U(t) on Q[i - 1] and Q[i].
* if k[j] = 1, apply X on Q[j] else, apply Z on Q[j].
Measure quantum register Q and store it in the classical register (304).
Repeat step 2 and 3 for S number of times and store the counts for each repeat in the form of a dictionary given by where = 1 and each xi refers to the frequency of appearance of the states in the set , respectively.
Find the output state |?? ? {|0?,|1?}N corresponding to the maximum of set {x1,x2,...,x2N}, convert |?? into a binary string ? ? {0,1}N, this string is the ciphertext of input message m.
Generation of input-output maps:
Parameters
Key space: Set K = {k|k ? {0,1}L}.
Message space: Set M = {m|m ? {0,1}n}.
Let ?(i,j) denote the n bit binary cipher text for key k and input message m.
Algorithm:
1. For i ? [1,2L]:
Select ki from the set K.
For j ? [1,2n]:
Pick mj from the set M.
Execute the algorithm GPRS(U,ki,mj) = ?i,j.
Store the above result in the dictionary as the following,
…………(12)
Store the above dictionary Fi for all possible key in key space K in the form of list as follows,
……………..(13)
Checking one-to-one and reversibility of input-output maps.
Algorithm:
1. For i ? [1,2L]:
Compare the number of elements in the dictionary NFi for ?i and number of elements in the set N?i for { ?i,j | j = 1,2,3,...,2n }.
If NFi = N?i , include the dictionary Fi as F’i .
If NFi ? N?i , discard the dictionary Fi .
Let the new list of dictionaries with length a be denoted by,
……………….(14)
Iterate through the dictionary F’PRS with loop variable i.
Compute the GPRS(U† , ki , ?j) = mi,j for each .
If GPRS( U†, Ki ,?j ) = GPRS( U, ki , mj), then include, else discard.
Collision-less and reversible keyed-transformations for N, L = 2, 4, 6, 8 bits.:
In an exemplary embodiment, the system (102) checks a test case by taking the unitary as N = 4, Q = 2 SYK Hamiltonian and the parameters as, p = -106, k = 100, t = p, and the J coupling coefficient list given by, [ ]. The time evolution for each circuit is considered t = p seconds. In applied form, the total Hamiltonian H is of the form,
……………(15)
The Hamiltonian and unitary operators for N = 2 ,4 ,6 and 8 can be derived as follows,
………….(16)
The Hamiltonian HSY K is given by the following equation,
HSYK = (159139.02759758616) X ? I + (159170.85858620453) Y ? Z - (1000000.0) Y ? Y + (1000000.0) Z ? Z + (159170.85858620453) Z ? Y + (159139.02759758616) I ? X.
Corresponding inverse unitaries are given by,

Figures 5a-5d illustrate schematic diagrams (500) depicting forward quantum circuits for N, L = 2, 4, 6, 8 bits, in accordance with an exemplary embodiment of the present disclosure. In Figures 5a-5d, two qubit Hamiltonian, unitary and unitary conjugate transpose operator matrix form for N4Q2 SYK model.
Figure 5a illustrates a forward quantum circuit which may be a 2-qubit forward quantum circuit for N = L = 2, and with m = 00 and K = 11.
Figure 5b illustrates a forward quantum circuit which may be a 4-qubit forward quantum circuit for N = L = 4, and with m = 0000 and K = 1111.
Figure 5c illustrates a forward quantum circuit which may be a 6-qubit forward quantum circuit for N = L = 6, and with m = 000000 and K = 111111.
Figure 5d illustrates a forward quantum circuit which may be a 8-qubit forward quantum circuit for N = L = 8, and with m = 00000000 and K = 11111111.
In an embodiment, the system (102) may be configured to generate a forward input-output Table for forward quantum circuits for N, L = 2, 4, 6, 8 bits N, L = 2, 4, 6, 8 bits. Table 1 illustrates 4 × 4 elements forward input-output Table for N,L = 2 bits. Table 2 illustrates 16 × 16 elements forward input-output Table for N, L = 4 bits. Table 3 illustrates 1 × 64 elements forward input-output Table for N, L = 6 bits and key K = 111111. Table 4 illustrates 1 × 256 Elements input output Table for N,L = 8. Table 5 illustrates a 1 × 256 Elements input output Table for 4 - 7 for N,L = 8. Table 6 illustrates 1 × 256 Elements input output Table for 8 - 11 for N,L = 8. Table 7 illustrates 1 × 256 Elements input output Table for 12 - 15 for N,L = 8.
Figure 6a illustrates an inverse quantum circuit which may be a 2-qubit inverse quantum circuit for N = L = 2, and with m = 00 and K = 11.
Figure 6b illustrates an inverse quantum circuit which may be a 4-qubit inverse quantum circuit for N = L = 4, and with m = 0000 and K = 1111.
Figure 6c illustrates an inverse quantum circuit which may be a 6-qubit inverse quantum circuit for N = L = 6, and with m = 000000 and K = 111111.
Figure 6d illustrates an inverse quantum circuit which may be a 8-qubit inverse quantum circuit for N = L = 8, and with m = 00000000 and K = 11111111.
In an embodiment, the system (102) may be configured to generate a reverse output-input tables for inverse quantum circuits for N, L = 2, 4, 6, 8 bits N, L = 2, 4, 6, 8 bits. Table 8 illustrates 4 × 4 elements reverse output-input Table for N, L = 2 bits. Table 9 illustrates 16 × 16 elements reverse output-input Table for N, L = 4 bits. Table 10 illustrates 1 × 64 elements reverse output-input Table for N, L = 6 bits and Key K = 111111. Table 11 illustrates 1 × 256 Elements reverse output-input Table 0 - 3 for N,L = 8. Table 12 illustrates 1 × 256 Elements reverse output-input Table 4 - 7 for N,L = 8. Table 13 illustrates 1 × 256 Elements reverse output-input Table 8 - 11 for N,L = 8. Table 14 illustrates 1 × 256 Elements reverse output-input Table 12 - 15 for N,L = 8.
Figure 7 illustrates a schematic diagram (700) depicting a quantum circuit for hardware implementation, in accordance with an exemplary embodiment of the present disclosure.
In an exemplary embodiment, Figure 7 illustrates an Experimental implementation of a quantum circuit schematic for N, L = 2 classical bit input-output and key. In this, experimental implementation of the AdS/CFT shock operators has been provided for N,L = 2. The quantum circuit corresponding input message = ’11’ and input key = ’01’ is given along with the matrix form of the unitary operator. Table 15 illustrates a matrix form of unitary U used in the quantum circuit.
Figure 8 illustrates a schematic diagram (800) depicting a decomposed quantum circuit corresponding to a unitary operator, in accordance with an exemplary embodiment of the present disclosure.
In an exemplary embodiment, the system (102) may employ a universal gate-model QPU called the ”Rigetti—Aspen-M-2” quantum computer, which is based on superconducting qubits. Rigetti quantum computer is provided by Amazon Bracket and has the particular set of quantum gates which can be used for the experimental implementation. So, the system (102) first decomposes the above quantum circuit in terms of quantum gates which are available in Rigetti quantum computer.
Other Experimental implementation parameters:
Number of qubits: 2
Number of shots: 1000
Applications in cryptography:
PRS are efficiently computable quantum states with pseudorandom nature at its core, i.e., indistinguishable from Haar random states. PRS can used in various cryptographic applications as well as in several other areas such as quantum money. PRS also finds its application in multi-party computation and zero knowledge proofs where it can be used as a primitive for commitment schemes. In private key cryptography, PRS can be used to replace primitives which requires keyed transformations. PRS can be extended to get pseudorandom function like states (PRFS). This can be done using key splitting technique. The PRFS can further be used to build an encryption scheme similar to one time pad which will be more secure as compared to the classical encryption schemes. Alternatively, PRS can act as a quantum analogue of Pseudorandom Generator (PRG) lacking the capability to stretch or/and shrink output. PRFS, a generalization of PRS can be obtained using the proposed technique for PRS generation. PRFS can effectively replace the usage of PRGs and PRFs in cryptographic primitive like Naor commitment. In the classical version of Naor commitment, the pseudorandom generator G is used to map a ? bits input to 3? bits, which looks random. Similarly in the quantum analogue, a (d,n = 7?)-PRFS with input k can be constructed from any n-qubit PRS. This PRFS can be implemented in commitment scheme in the place of PRG, which looks like 2d Haar random states.
Conditions for encryption scheme: The system (102) may provide the possible application in the encryption schemes. The system (102) provides the required conditions for the SYK shock circuit based PRS permutations to act as encryption-decryption mechanisms.
Deterministic/one-to-one constraint on N = 4, Q = 2 SYK Hamiltonian operator: The Hamiltonian operator for N = 4 and q = 2 SYK model is given by,
………….(15)
with Jij are random coefficients following gaussian distribution with Jij = 0.
For N = 4 and q = 2, we will have 4C2 = 6 different combinations of ?i?j. Thus, the Hamiltonian after applying Jorden-Wigner transformation is as follows:
……….(16)
with = I. Now, if the system (102) wants the corresponding 22 ×22 Hamiltonian HSY K and Unitary Operator USY K = eiHSYKt to generate unique maximum probability peaks, then the system (102) ensures that for some special coupling coefficients and time evolution t*, the quantum state evolution obeys the following rule,
……….(17)
where , such that, for some aj , j ? {1,2,3,4} the following condition holds,
………. (18)
Similar conditions should follow for higher . Here, the final probabilities Pj = |aj|2 can be written as,

where pi = probabilities of the two qubit outputs.
The maximum value of Pj would correspond to,

Here, if , then . For, n = 128, 0.52/n ˜ 0.989 and consider that Max(pi) > 0.989.
Restriction on measurement-existence of the max function: Supposing the condition given in equation (18) holds true, if the system (102) applies the Pauli Z measurement operator on N = 2 qubits, then measurement M operator is given by,

……..(19)
Then, probability of measuring the states |00?,|01?,|10?,|11? is given by, (19)
…….. (20)

The expected probabilities would be, (21)
(22)
(23)
(24)
(25)
If the system (102) applies a Max() function over the set p = {P1,P2,P3,P4}, then it is supposed to return a Pj ? p such that,
where
Overall constraint on SYK shock circuit - shuffled permutations of the rows and columns of identity matrix: The quantum algorithm can be simply described by the following equation, keeping an arbitrary initial state |?? ? {|0?,|1?}N via,
………….(26)
where, each and Oi ? {I, X, Y, Z} and the shock operator set {Oi} is determined by a key K ? {0,1}L=N. If the system (102) does not have the condition satisfied then, the system (102) can impose the condition on Ushock = UlOlUl-1Ol-1...O1U1 to be a 2N × 2N matrix which is a permutation of the rows and columns of an identity matrix of the same dimensions, given by,
…….(27)
Figure 9 illustrates a block diagram (900) depicting a SYK shock circuit PRS generator-based quantum Merkle tree, in accordance with an embodiment of the present disclosure.
In an embodiment, the system (102) may be configured to provide a complete quantum Merkle Tree implementation wherein all the layers (904, 906, 908) are implemented via a N4Q2 SYK Shock Circuit PRS Generation Model.
In an embodiment, the system (102) also includes a SYK Shock Circuit PRS Generator based Quantum Merkle Tree (901). The system (102) provides a N4Q2 SYK PRS generator for a Quantum Merkle Tree realization.
In an exemplary embodiment, the experimental result of the system (102) are as follows.
The system (102) first takes an input (902) which contains 8 x 2-bit input transaction information (labelled as T0, T1, T2, T3, T4, T5, T6, T7) given as inputs to a SYK Shock Circuit (905), which is as acting as a 2N to N bit PRS permutation and compression mechanism via a quantum circuit, this is termed as the first layer (904), leading to generation of 4 binary Hash values given as {#01, #23, #45, #67}, which act as the input to the second SYK Shock Circuit layer (906) yielding {#0123, #4567}, these are processed by the final layer (908) to generate the Quantum Merkle root {#01234567} (910).

, Claims:We claim:
1. A method for generating pseudorandom quantum states (PRS) using the SYK Hamiltonian and the AdS/CFT correspondence, comprising:
encoding a binary input message onto a quantum register;
applying N/2 pairs of 2-qubit SYK Unitaries (U_SYK) in parallel on the quantum register;
applying shock operators alternately on the qubits based on a secret key to generate permutations;
performing measurements in the computational basis to estimate state amplitudes;
selecting an outcome with maximum probability as the output of a quantum circuit;
generating an input-output map for a fixed key;
Using the input-output map to generate permutations for cryptographic applications.
2. The method as claimed in claim 1, further comprising limiting the length of the secret key to the number of qubits in the quantum circuit.
3. The method as claimed in claim 1, wherein the SYK Hamiltonian includes M terms, each belonging to an N-dimensional Hilbert space, and wherein the SYK Hamiltonian is configured with specific coupling coefficients, J, for enhanced security and randomness in the generated states.
4. The method as claimed in claim 1, wherein the binary input message is encoded onto the quantum register by applying X gates for binary values of 1 and I gates for binary values of 0.
5. The method as claimed in claim 1, wherein the input-output map is generated by running the quantum circuit a fixed number of times for every input state, ensuring accuracy in amplitude estimation.
6. The method as claimed in claim 1, wherein the 2-qubit SYK Unitaries (U_SYK) are applied using the Suzuki Trotter method for approximating the time evolution operator.
7. A method for generating pseudorandom quantum states, comprising:
encoding a binary input message onto a quantum register using X gates for binary 1 and I gates for binary 0;
applying 2-qubit SYK Unitaries (U_SYK) using the Suzuki Trotter method for approximating the time evolution operator;
utilizing shock operators in a predetermined sequence based on a secret key to perturb the quantum states;
running the quantum circuit a fixed number of times for every input state to create an input-output map, and
generating permutations for cryptographic applications, including S-Boxes for encryption/decryption circuits.

8. A system for generating pseudorandom quantum states, comprising:
a quantum circuit processor with a quantum register for encoding binary input messages, applying 2-qubit SYK Unitaries, and utilizing shock operators based on a secret key;
a classical processing/storage device with a classical register for storing measurement results obtained from the quantum circuit, and
a permutation generation module configured to create input-output maps by running the quantum circuit multiple times, wherein the system configured for generating permutations for cryptographic applications, including S-Boxes for encryption/decryption circuits.