Description: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 of a system and method which processes the quantum computing techniques for generating permutations to overcome the above limitations.
SUMMARY
A method for generating pseudorandom quantum states (PRS) using quantum computing techniques is provided. The method includes applying pairs of quantum unitaries in parallel on a quantum register, each unitary defined as a mathematical transformation involving quantum gates; employing shock operator sequences alternately during unitary application, guided by a key determining the sequence of shock operators. The method further includes performing a measurement in a computational basis to obtain a distribution, estimating amplitudes of the generated state; selecting an outcome with maximum probability as the output of a quantum circuit, and iteratively running the circuit for each input state, providing a distribution for accurate amplitude estimation.
In another embodiment, a method for generating pseudorandom quantum states (PRS) using the SYK Hamiltonian and AdS/CFT correspondence is disclosed. The method includes applying N/2 pairs of 2-qubit SYK Unitaries (U_SYK) in parallel on a quantum register, each U_SYK defined as exp(iH_SYKt), where 'exp' is an exponential function, 'i' is imaginary, 'H_SYK' is the 2-qubit SYK Hamiltonian, and 't' is a time evolution value. The method further includes applying N shock operator sequences (O1, O2, ..., O(N-1), O(N)) alternately during SYK Unitary application, governed by a fixed key (K(i)) that determines the sequence of shock operators, and performing a measurement in a computational basis to obtain a distribution, estimating amplitudes of the generated state. Further, the method includes selecting an outcome with maximum probability as the output of a quantum circuit, and iteratively running the circuit a fixed number of times for each input state, providing a distribution for accurate amplitude estimation.
In another embodiment, determining an input-output map for a fixed key, wherein the key determines the sequence of shock operators to be applied, and each input state consistently maps to the same output with maximum probability.
. In another embodiment, the distribution is determined by running the circuit a fixed number of times for every input state, and a larger number of iterations improves the accuracy of amplitude estimation.
In another embodiment, wherein the input-output map generated for specific values of parameters in the time evolution unitary provides a permutation, applicable for cryptographic protocols as potential S-boxes.
In another embodiment, wherein the generated permutations for cryptographic purposes, establishing a groundwork for potential applications of the AdS/CFT conjecture in cryptography
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.
The present disclosure is described in detail with reference to the accompanying figures. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. The same numbers are used throughout the drawings to refer various features of the present subject matter.
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.
The figures depict various embodiments of the present invention for purposes of illustration only. One skilled in the art will readily recognize from the following discussion that alternative embodiments of the structures and methods illustrated herein may be employed without departing from the principles of the invention described herein.
DETAILED DESCRIPTION
Some embodiments of this disclosure, illustrating all its features, will now be discussed in detail. The words "comprising," "having," "containing," and "including," and other forms thereof, are intended to be equivalent in meaning and be open ended in that an item or items following any one of these words is not meant to be an exhaustive listing of such item or items or meant to be limited to only the listed item or items. It must also be noted that as used herein and in the appended claims, the singular forms "a," "an," and "the" include plural references unless the context clearly dictates otherwise. Although any computing platforms and methods, similar or equivalent to those described herein can be used in the practice or testing of embodiments of the present disclosure, the exemplary, computing platforms and methods are now described. The disclosed embodiments for processing quantum computing techniques are merely examples of the disclosure, which may be embodied in various forms.
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.
In an embodiment, 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.
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 system (102) may comprise the cloud-based computing environment in which a user, interchangeably may referred to as a consumer, may operate individual computing systems configured to execute remotely located applications. Examples of the user devices (104) may include, but are not limited to, a portable computer, a personal digital assistant, a handheld device, and a workstation. The user devices (104) are communicatively coupled to the system (102), a database (108) through a network (106).
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).
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. The detailed description of the modules (208) along with other components of the system (102) is further explained by referring to Figure 3 and Figure 4.
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). 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 anti-de Sitter ”spacetime” has a boundary where the conformal field theory lies. The AdS/CFT correspondence is basically a duality between two theories living in two different dimensions, justifying its nickname as “Holography”. In the sense that there is a “dictionary” for transforming calculations in one theory into calculations in the other, it is asserted that this conformal field theory is analogous to the gravitational theory on the bulk anti-de Sitter space. Each element in one theory has an equivalent in the other.
The Sachdev-Ye-Kitaev (SYK) model is an exactly solvable nearly conformally invariant model for which describes a strongly coupled, chaotic quantum many-body system. The discrete AdS/CFT model and fermionic code are closely connected to the model, which is thought to shed light on the understanding of highly correlated materials. The 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. This can be formally proved to be pseudorandom if it can satisfy the two conditions of pseudorandomness.
There is a need to show that these states can be generated efficiently in a polynomial time by a quantum algorithm when provided with the key as input. This part can be easily proven by showing that the circuit depth is polynomial. A quantum circuit (306) consists of alternating SYK Hamiltonian time evolution and the shock operators. The time evolution operator is a unitary operator and can thus be efficiently simulated on a quantum computer by using a simulation module (214) in polynomial complexity w.r.t the number of qubits. Furthermore, the shock operators which are just Pauli operators acting on a single qubit can be simulated in poly time on a quantum computer. The total circuit depth will depend on the number of input qubits and the number of shock operators applied to the circuit. The state preparation would thus be O (l × tscr) which means that it can be generated efficiently by any quantum computer once they know the shock operators to apply.
The ensemble of states generated after the SYK Hamiltonian with shock operators must be indistinguishable from a Haar random ensemble of states for any poly time algorithm acting on polynomial copies of the state. This can be proven by showing that the distinguishing algorithm must take exponential time or must be provided with exponentially many copies of the states. The distinguishing algorithm makes use of the search algorithm to go over the entire ensemble to determine whether the state belongs to the pseudorandom ensemble or a Haar ensemble. Previously, it showed that the number of possible states, i.e., the size of the pseudorandom ensemble depends exponentially on the number of shock operators applied. For a circuit with l shock operators, we will have an ensemble of 2l states. The distinguisher algorithm must be at least as complex as a search technique. A given quantum search technique would be square root with the number of queries. Here, the system (102) includes 2l states and thus the distinguisher would be ?(2l/2) complex. Thus, it would at least take exponential time or exponential copies to accurately distinguish between a Haar state and a pseudorandom state generated by the SYK Hamiltonian with shock operator method.
Here, it is shown that the states obtained using the SYK Hamiltonian with shock operators are indeed pseudorandom states. These states can now be used to generate keyed transformations, where the sequence of shock operators can act as the key.
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.
Constructing Shock Operators from Key Sequence:
The CFT state corresponding to the shock operators is defined as:
………….(10)
where, Ui = e-i HCFT tl and Oi picked randomly from the Pauli Operators set {X,Y,Z,I} according to some Uniform Random Distribution P. Then the shock operator sequence will be of the form {Ol,Ol-1,..O0} with dim(|?shock?) = 2N × 1 where N = total qubit number. Now, from the definition of PRS, the system (102) compares the shock operator sequence with a key K ? {0,1}? with ? < N from above. Suppose, the system (102) defines three distinct sets SX = {I, X}, SY = {I, Y} and SZ = {I, Z}, now corresponding to K ? {0, 1}? and keeping ? = l, the system (102) defines shock operator sequences OX ? {I, X}l, OY ? {I, Y}l and OZ? {I, Z}l respectively. In general, form these sets can be described as implying si = X,Y,Z respectively. The system (102) can also define a set of mixed shock operators, for example, in the case of N = 2, l = 5, the sequence can also be of the form,
………….(11)
but with the restriction that only random Pauli operator is applied when a shock operator is implemented.
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.
Time Complexity Analysis: In an embodiment, the system (102) provides three scenarios for execution of the SYK Quantum Merkle Tree (900), and analyses the time complexity in terms of circuit width, depth and processing time. The relevant notation is as follows, n? = length of input message, d? = circuit depth, N = number of transactions.
Scenario 1: Simulation of single branch.

Scenario 2: Simulation of an individual layer of multiple branches.

Scenario 3: Simulation of an entire Quantum Merkle Tree.

Implementation on Quantum Hardware: In an embodiment, the system (102) may be configured to implement the scenario 1 from the above section on AWS Rigetti Aspen-M-3 (911) and provide a detailed schematic and experiment results from the quantum hardware implementation.
Table 17 illustrates the results from the above implementation.
Layer and Branch Value Inputs (binary) Generated Hash Value (binary) Processing Time
(seconds)
Layer 1: Branch 1 (T0 = 11, T1 = 00) #01 = 11 9.001
Layer 1: Branch 2 (T2 = 00, T3 = 00) #23 = 00
10.890
Layer 1: Branch 3 (T4 = 00, T5 = 00)
#45 = 00
9.273
Layer 1: Branch 4 (T6 = 10, T7 = 11)
#67 = 10
9.276
Layer 2: Branch 1
(#01, #23)
#0123 = 11 8.835
Layer 2: Branch 2
(#45, #67) #4567 = 01 8.847
Layer 3: Quantum Merkle Root
(#0123, #4567) #01234567 = 01 9.893
Table 17: Results of Quantum Hardware
It should be noted that the description merely illustrates the principles of the present inventions. It will thus be appreciated that those skilled in the art will be able to devise various arrangements that, although not explicitly described herein, embody the principles of the present invention. Furthermore, all examples recited herein are principally intended expressly to be only for explanatory purposes to help the reader in understanding the principles of the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Moreover, all statements herein reciting principles, aspects, and embodiments of the invention, as well as specific examples thereof, are intended to encompass equivalents thereof.
, Claims:We claim:
1. A method for generating pseudorandom quantum states (PRS) using quantum computing techniques, comprising:
applying pairs of quantum unitaries in parallel on a quantum register, each unitary defined as a mathematical transformation involving quantum gates;
employing shock operator sequences alternately during unitary application, guided by a key determining the sequence of shock operators;
performing a measurement in a computational basis to obtain a distribution, estimating amplitudes of the generated state;
selecting an outcome with maximum probability as the output of a quantum circuit, and
iteratively running the circuit for each input state, providing a distribution for accurate amplitude estimation.

2. The method as claimed in claim 1, further comprising determining an input-output map for a fixed key, wherein the key determines the sequence of shock operators to be applied, and each input state consistently maps to the same output with maximum probability.

3. The method as claimed in claim 1, wherein the distribution is determined by running the circuit a fixed number of times for every input state, and a larger number of iterations improves the accuracy of amplitude estimation.

4. The method as claimed in claim 1, wherein the input-output map generated for specific values of parameters in the time evolution unitary provides a permutation, applicable for cryptographic protocols as potential S-boxes.

5. The method as claimed in claim 1, further comprising utilizing the generated permutations for cryptographic purposes, establishing a groundwork for potential applications of the AdS/CFT conjecture in cryptography.

6. The method as claimed in claim 1, incorporating a quantum circuit processor with a quantum register for initial state configuration and a classical processing/storage device for storing measurement results and applying measurement operators in the Pauli Z basis.

7. The method as claimed in claim 1, simulating the quantum circuit on a classical processing/storage device using an Aer Qiskit Simulator as part of a simulation module, or implementing quantum hardware such as an AWS Rigetti Quantum Hardware for the quantum circuit.

8. The method as claimed in claim 1, leveraging the AdS/CFT correspondence to enhance the generation of PRS, utilizing the duality between Anti-de Sitter spaces (AdS) and Conformal field theories (CFT) for improved comprehension of quantum gravity and string theory.

9. A method for generating pseudorandom quantum states (PRS) using the SYK Hamiltonian and AdS/CFT correspondence, comprising:
applying N/2 pairs of 2-qubit SYK Unitaries (U_SYK) in parallel on a quantum register, each U_SYK defined as exp(iH_SYKt), where 'exp' is an exponential function, 'i' is imaginary, 'H_SYK' is the 2-qubit SYK Hamiltonian, and 't' is a time evolution value;
applying N shock operator sequences (O1, O2, ..., O(N-1), O(N)) alternately during SYK Unitary application, governed by a fixed key (K(i)) that determines the sequence of shock operators;
performing a measurement in a computational basis to obtain a distribution, estimating amplitudes of the generated state;
selecting an outcome with maximum probability as the output of a quantum circuit, and
iteratively running the circuit a fixed number of times for each input state, providing a distribution for accurate amplitude estimation.

10. The method as claimed in claim 9, simulating the quantum circuit on a classical processing/storage device using a quantum computing simulator as part of a simulation module, or implementing quantum hardware for the quantum circuit.