Cumulative Memory Lower Bounds for Randomized and Quantum Computation

A novel technique for obtaining lower bounds for the time versus space complexity of certain functions in a general input oblivious sequential model of computation is developed. This is demonstrated by studying the intrinsic complexity of the following set equality problem SE(n,m): Given a sequence x1,x2,....,xn, y1,....,yn of 2n numbers of m bits each, decide whether the sets [x1,....,xn] and [y1,...,yn] coincide. We show that for any log log n ≤ m ≤1/2log n and any 1 ≤ s ≤ log n, any input oblivious sequential computation that solves SE(n,m) using 2m/s space, takes Ω(n ? s) time. This result is sharp for all admissible values of n,m,s and is the first known nontrivial time space tradeoff lower bound (for space = ω (log n) of a set recognition problem on such a general model of computation. Our method also supplies lower bounds on the length of arbitrary (not necessarily input oblivious) branching programs for several natural symmetric functions, improving results of Chandra, Furst and Lipton, of Pudlák and of Ajtai et. al. For example we show that for the majority - function any branching program of width w(n) has length ω(n · log w/n (n) · log w (n)), in particular for bounded width we get length ω (n log n) (independently of our work Babai et. al. [BPRS] have simultaneously proved this last result). Our lower bounds for branching programs imply lower bounds on the number of steps that are needed to pebble arbitrary computation graphs for the same computational problems. To establish our lower bounds we introduce the new concept of a meander that captures superconcentrator-type properties of sequences. We prove lower bounds on the length of meanders via a new Ramsey theoretic lemma that is of interest in its own right. This lemma has other applications, including a tight lower bound on the size of weak superconcentrators of depth 2 that strengthens the known lower bound of Pippenger [Pi]. A surprising new feature of these applications of Ramsey theory in lower bound arguments is the fact that no numbers are required to be unusually large and that several of the resulting superlinear lower bounds are in fact optimal.

Cryptography is a practice of technique to ensure security by using the cryptography keys. Key derivation function (KDF) is a standard algorithm to generate these cryptographic keys. Stream ciphers are one of the cryptographic primitives that are used to construct the key derivation function namely key derivation function based on stream ciphers. Though the key derivation function based on stream ciphers have a great role in security, it is necessary to have a framework which can evaluate the security level of the different types of key derivation function based on stream ciphers. Random oracle model (ROM) is the current procedure to proofs the security of KDF. However, the security evaluation of ROM did not evaluate the degree of secureness of KDF as it can only proof either the KDF is theoretically secure or insecure. Hence, this research applies fuzzy evaluation method to form a framework to evaluate the degree of secureness of the KDF for different types of key derivation function based on stream ciphers. Key sizes and complexity attacks are two main variables which are considered in the design of fuzzy rule. The proposed method introduces the information extraction to construct fuzzy membership function and rules. The result from this proposal is effective to approximate the security aspect in the computer system as well as network system.

In this paper, we clarify the security of practical cryptosystems with hash functions based on key derivation functions (KDFs). We use the indifferentiability framework in order to discuss the security because the indifferentiability from Random Oracle (and its variants) guarantees that cryptosystems remain secure even if Random Oracles (ROs) are instantiated with hash functions. Though previous works on the indifferentiability of Merkle-Damgård (MD) hash functions focus on stand-alone hash functions, there is no work which focuses on MD hash functions with KDFs. Many cryptosystems need longer output lengths of hash functions than stand-alone hash functions and KDFs are used to generate longer digests as specified in PKCS #1 v2.1 and IEEE P1363. Specifically, we obtain the following results. We denote the MD hash function using Stam’s type-II compression function by MD-SCFII and MD-SCFII with KDFs by KDF-MD-SCFII. Cryptosystems secure in the pub-RO model (FDH, PSS, Fiat-Shamir, and so on): Dodis et al. proposed the indifferentiability from pub-RO to prove the security of these cryptosystems using MD-SCFII while did not consider the KDF structures. So we propose a different framework, indifferentiability from privleak-RO. Using this framework and their result, we show that these cryptosystems using KDF-MD-SCFIIs are secure. Encryption schemes secure in the RO model (OAEP, RSA-KEM, PSEC-KEM, ECIES-KEM and so on): The encryption schemes are secure in the “fixed inputl length” RO model because the input lengths of ROs from the encryption schemes are fixed. We show that this fact guarantees the security of the encryption schemes using KDF-MD-SCFII. KeywordsHash FunctionEncryption SchemeRandom OracleCompression FunctionRandom Oracle ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents a comprehensive formal security framework for key derivation functions (KDF). The major security goal for a KDF is to produce cryptographic keys from a private seed value where the derived cryptographic keys are indistinguishable from random binary strings. We form a framework of five security models for KDFs. This consists of four security models that we propose: Known Public Inputs Attack (KPM, KPS), Adaptive Chosen Context Information Attack (CCM) and Adaptive Chosen Public Inputs Attack(CPM); and another security model, previously defined by Krawczyk [6], which we refer to as Adaptive Chosen Context Information Attack(CCS). These security models are simulated using an indistinguisibility game. In addition we prove the relationships between these five security models and analyse KDFs using the framework (in the random oracle model).

In order to improve the training efficiency of the support vector machine, a quantum circuit training scheme based on the inner product of the quantum state for the support vector machine is proposed in this work. Firstly, on the basis of the full analysis of the computational complexity of the classical support vector machine, the kernel function which is the main factor affecting the computational complexity of the algorithm is primarily analyzed. Based on quantum mechanics and quantum computing theory, the training sample elements in the kernel function are quantized to generate the corresponding quantum states. Secondly, according to the quantum states of the training sample elements, the types and quantities of the required quantum logic gates are derived and calculated, and the quantum circuit that can generate the corresponding quantum states of the training sample elements through the evolution of the quantum initial ground states and the quantum logic gates is designed. Then, in the light of the relationship between the inner product of the quantum state and the quantum logic gate SWAP, the quantum circuit is designed to complete the exchange operation of the corresponding quantum state amplitude. The inner product of the quantum state is realized by exchanging and evolving the amplitude of the quantum state in the quantum circuit. Finally, by measuring the quantum state of the controlling qubit, the inner product solution of the kernel function is obtained, and the acceleration effect of training support vector machine is realized. The verification results show that the scheme enables the support vector machine not only to complete the correct classification, but also to operate the quantum part of the scheme on the real quantum computer . Compared with the classical algorithm, the scheme reduces the time complexity of the algorithm for the polynomial degree, greatly shortens the training time of the model, and improves the efficiency of the algorithm. The scheme has certain feasibility, effectiveness and innovation, and expands the training idea of the support vector machine.

Cryptographic Primitives that Resist Backdooring and Subversion

Scaling quantum computers, i.e., quantum processing units (QPUs) to enable the execution of large quantum circuits is a major challenge, especially for applications that should provide a quantum advantage over classical algorithms. One approach to scale QPUs is to connect multiple machines through quantum and classical channels to form clusters or even quantum networks. Using this paradigm, several smaller QPUs can collectively execute circuits that each would not be able on its own. However, communication between QPUs is costly as it requires generating and maintaining entanglement, and hence it should be used wisely. In this paper, we evaluate the architectures, and in particular the entanglement patterns, of variational quantum circuits in a distributed quantum computing (DQC) setting. That is, using Qiskit, we simulate the execution of an eight qubit circuit using four QPUs each with two computational and two communication qubits where non-local CX-gates are performed using the remote-CX protocol. We compare the performance of various circuits on a binary classification task where training is executed under ideal and testing under noisy conditions. The study provides initial results on suitable VQC architectures for the DQC paradigm, and indicates that a standard VQC baseline is not always the best choice, and alternative architectures that use entanglement between QPUs sparingly deliver better results under noise.

Recent improvements in control of quantum systems make it seem feasible to finally build a quantum computer within a decade. While it has been shown that such a quantum computer can in principle solve certain small electronic structure problems and idealized model Hamiltonians, the highly relevant problem of directly solving a complex correlated material appears to require a prohibitive amount of resources. Here, we show that by using a hybrid quantum-classical algorithm that incorporates the power of a small quantum computer into a framework of classical embedding algorithms, the electronic structure of complex correlated materials can be efficiently tackled using a quantum computer. In our approach, the quantum computer solves a small effective quantum impurity problem that is self-consistently determined via a feedback loop between the quantum and classical computation. Use of a quantum computer enables much larger and more accurate simulations than with any known classical algorithm, and will allow many open questions in quantum materials to be resolved once a small quantum computer with around one hundred logical qubits becomes available.

Simulation of the dynamics of quantum materials is emerging as a promising scientific application for noisy intermediate-scale quantum (NISQ) computers. Due to their high gate-error rates and short decoherence times, however, NISQ computers can only produce high-fidelity results for those quantum circuits smaller than some given circuit size. Dynamic simulations, therefore, pose a challenge as current algorithms produce circuits that grow in size with each subsequent time-step of the simulation. This underscores the crucial role of quantum circuit compilers to produce executable quantum circuits of minimal size, thereby maximizing the range of physical phenomena that can be studied within the NISQ fidelity budget. Here, we present two domain-specific (DS) quantum circuit compilers for the Rigetti and IBM quantum computers, specifically designed to compile circuits simulating dynamics under a special class of time-dependent Hamiltonians. The compilers outperform state-of-the-art general-purpose compilers in terms of circuit size reduction by around 25%–30% as well as wall-clock compilation time by around 40% (dependent on system size and simulation time-step). Drawing on heuristic techniques commonly used in artificial intelligence, both compilers scale well with simulation time-step and system size. Code for both compilers is open-source and packaged into a full-stack quantum simulation software with tutorials included for ease of use for future researchers wishing to perform dynamic simulations of quantum materials on quantum computers. As our DS compilers provide significant improvements in both compilation time and simulation fidelity, they provide a building block for accelerating progress toward physical quantum supremacy.

While current quantum computers, referred to as Noisy Intermediate-Scale Quantum (NISQ) computers, are expected to be beneficial for different applications, they are prone to different types of errors. In order to enhance the reliability of quantum systems, noise-aware quantum compilers are used to generate physical quantum circuits to be executed on NISQ computers. The quantum hardware is calibrated very frequently and its error rates are computed accordingly. Based on the hardware error rates, a quantum compiler allocates physical qubits and schedules quantum operations. However, error rates may change post-calibration. To incorporate dynamic error rates into quantum circuit compilation with minimum cost, we propose a Machine Learning (ML)-based scheme to detect the incorrect output of the quantum circuit and predict the Probability of Successful Trials (PST) with high accuracy. Our approach can verify the error rates of the quantum hardware and validate the correctness of the extracted quantum circuit output. We provide a case study of our ML-based reliability models using IBM Q16 Melbourne quantum computer. Our results show that the proposed scheme achieves a very high prediction accuracy.

Quantum computing simulation platform can simulate the computation results of the quantum computer based on traditional computers, which is an effective way to promote the development of quantum computing software, algorithms and hardware at the current immature stage of the real quantum computer. Since quantum computers have exponential calculation acceleration compared with traditional computers, the main problems in implementing quantum computing simulation on traditional computers are low computational efficiency and long time-consuming. A quantum circuit which is a sequence of quantum gates acting on a collection of qubits is the general quantum computing model. So by the means of quantum circuit optimization, the calculation speed can be significantly increased while keeping the calculation result unchanged. The existing empirical rules of quantum circuit optimization methods have limitations and there is no common and automatic quantum circuit optimization method. In this paper, a general and automatic quantum circuit optimization method based on the genetic algorithm is proposed, by which the equivalent optimal quantum circuit is obtained through a finite number of searching in a large searching space. This method is not limited by the hardware of the quantum computing simulation and the composition of the quantum circuit. The experimental results show that for the QFT algorithm of 29 qubits, the running time can be shortened by 41.4% and for the variational circuit of 6 qubits, the running time can be shortened by 18.8% compared with the state-of-the-art quantum circuit optimization method. So this method can improve the quantum computing simulation capability and operating efficiency and provide a rapid development way for quantum algorithms and applications.

Our aim in this paper is to present experimental evidence of a side-channel attack on superconducting cloud-based quantum computers, demonstrating that a quantum circuit can be analyzed indirectly using data gathered before and after its execution on a quantum computer. We hope that this can motivate further research into hardware and system stability of quantum computers as well as into more-nuanced systems-level attacks. Although much work has been done on establishing error correcting codes to address issues of decoherence, and schemes can be utilized to leverage quantum effects for the sake of intrusion detection over quantum communication channels, there is a lack of literature in the field regarding side-channel attacks on transpiled quantum circuits, particularly those which leverage quantum effects to gather information about quantum circuits running in the cloud. We explored issues of decoherence by examining quantum circuit behavior both before and after a specified circuit was executed in order to ascertain decalibration rates and potential sources of decoherence: our data clearly shows that information on an interim circuit can be acquired in this manner. Using convenient data classification techniques and various circuit diagrams consisting of one quantum gate and several measurements, we were able to distinguish between these various circuits when executed at position n in the interim phase by training our classifier on data from probing circuits executed at queue positions n-1 and n+1. This shows that there is sufficient information residing in the data before and after a circuit is executed on a cloud-based superconducting quantum computer to allow an attacker to ascertain information about an otherwise unknown third-party circuit and shows potential to form the basis of a side-channel attack in cases where queue positions can be examined to target a third party’s transpiled circuit during experimentation. This side channel attack merits further examination and extrapolation to identify more-complex circuits, and deeper statistical consideration to determine potential sources of information leakage and mitigation strategies.

Chapter 5 - Quantum Algorithms and Methods

This work contains the analysis of results received after running synthesized quantum circuits for training perceptron neural networks. The training is performed by creating a Grover's algorithm with a custom oracle function. The concept of synthesizing quantum circuits was showcased in the process of generating training circuits for three perceptron topologies, which were designed to test the accuracy of the synthesis process. The test circuits serve to prove that the proposed synthesis approach could be scaled to utilize more complex quantum computing systems and to solve more practical tasks. IBM's 100-qubit cloud quantum simulator was used as the debugging environment. Quantum circuits for described algorithms are generated by the "Naginata" quantum synthesizer, its source code is published and further documented on GitHub along with the code for the provided example algorithms. The article describes the processes behind the algorithm for synthesizing quantum circuits that perform the training process of single-layer perceptrons by finding their weights by filtering all possible input values through a predefined accuracy criterion. Since quantum computing is still in its early development phase, quantum circuits are created mainly by manual placement of logic elements. Implementing quantum algorithms, especially more use-case specific ones, directly on the quantum circuit level could lead to the circuit easily becoming too complex for human comprehension. Quantum Circuit Synthesizer "Naginata" was created to simplify the development and debugging process of quantum algorithms, by adding better clarity to their development process. In our case, better clarity for the development process is achieved by composing functions for commonly used operations performed in the implemented quantum algorithm. The programmer could now implement the quantum algorithm as a set of functions, instead of manually creating a circuit from single logic elements. After this, the synthesizer would handle the task of creating the data for placing logic elements on the circuit. This enables an opportunity of implementing quantum algorithms with higher-level commands. In the scope of this work, parametrically generated generic blocks for frequently used operations such as: the adder, multiplier and digital comparator were created and utilized to form the training circuits. The test results, proved that with the help of the proposed quantum synthesizer, these compositions could be used efficiently as building blocks for implementing quantum algorithms. And by visually comparing sizes of both code and circuit representations of the synthesized circuits, to the code examples used to synthesize these circuits, it is determined that the proposed approach for implementing quantum circuits greatly simplifies the processes of development and debugging a quantum algorithm.

The construction of quantum computers is based on the synthesis of low-cost quantum circuits. The quantum circuit of any Boolean function expressed in a Positive Polarity Reed-Muller (PPRM) expansion can be synthesized using Multiple-Control Toffoli (MCT) gates. This paper proposes two algorithms to construct a quantum circuit for any Boolean function expressed in a PPRM expansion. The Boolean function can be expressed with various algebraic forms, so there are different quantum circuits can be synthesized for the Boolean function based on its algebraic form. The proposed algorithms aim to map the MCT gates into the NCV gates for any quantum circuit by generating a simple algebraic form for the Boolean function. The first algorithm generates a special algebraic form for any Boolean function by rearrangement of terms of the Boolean function according to a predefined degree of term d term , then synthesizes the corresponding quantum circuit. The second algorithm applies the decomposition methods to decompose MCT circuit into its elementary gates followed by applying a set of simplification rules to simplify and optimize the synthesized quantum circuit. The proposed algorithms achieve a reduction in the quantum cost of synthesized quantum circuits when compared with relevant work in the literature. The proposed algorithms synthesize quantum circuits that can applied on IBM quantum computer.