Study of Decoherence in Quantum Computers: A Circuit-Design Perspective Abdullah Ash- Saki Mahabubul Alam Swaroop Ghosh Electrical Engineering Electrical Engineering Electrical Engineering Pennsylvania State University Pennsylvania State University Pennsylvania State University University Park, USA University Park, USA University Park, USA [email protected] [email protected] [email protected] Abstract—Decoherence of quantum states is a major hurdle interaction with environment [8]. The decoherence problem towards scalable and reliable quantum computing. Lower de- worsens with an increasing number of qubits. To mitigate coherence (i.e., higher fidelity) can alleviate the error correc- those, techniques like cryogenic cooling and error correction tion overhead and obviate the need for energy-intensive noise reduction techniques e.g., cryogenic cooling. In this paper, we code [9] are employed. However, these techniques are costly performed a noise-induced decoherence analysis of single and as cooling the qubits to ultra-low temperature (mili-Kelvin) re- multi-qubit quantum gates using physics-based simulations. The quires very high power (as much as 25kW) and error correction analysis indicates that (i) decoherence depends on the input requires many redundant physical qubits (one popular error state and the gate type. Larger number of j1i states worsen correction method named surface code incurs 18X overhead the decoherence; (ii) amplitude damping is more detrimental than phase damping; (iii) shorter depth implementation of a [10]). quantum function can achieve lower decoherence. Simulations Circuit level implications of decoherence are not well un- indicate 20% improvement in the fidelity of a quantum adder derstood. Therefore, a circuit level analysis of decoherence when realized using lower depth topology. The insights developed process is warranted to provide insights that may open avenues in this paper can be exploited by the circuit designer to choose the for more optimization and stabilization techniques and thus, right gates and logic implementation to optimize the system-level fidelity. enable the promise of quantum computers with the available Index Terms—Quantum computing; decoherence; fidelity. noisy-intermediate-scale quantum (NISQ) [11] technology. To this end, the following contributions are made in this paper: I. INTRODUCTION • We explore multi-qubit and multi-depth quantum circuits In 1982, Richard P. Feynman envisioned that to simulate from a circuit-design standpoint. nature we would need a quantum mechanical computer [1]. • We employ a Physics based Master equation framework Since then different industrial and academic groups have to model the dynamics of the qubit state with environ- worked diligently to realize a physical quantum computer. mental interaction (noise). The building block of the quantum computer is called a • We report a trend in decoherence with input pattern. quantum bit or qubit. Different groups have come up with • We show that lower-depth quantum circuits are better different physical realization of qubits including ion-trap [2], performing in terms of decoherence than higher-depth superconducting circuits [3], semiconductor quantum dots [4], circuits. a single atom in Silicon [5] etc. each with its own benefit The rest of the paper is organized as follows: in Section and caveat. Qubit possesses some unique properties like su- II, we discuss the basics of quantum computing and describe arXiv:1904.04323v1 [cs.ET] 8 Apr 2019 perposition, entanglement and quantum interference. Due to the simulation framework used in the paper. In Section III, these unique properties quantum computers are prophesied we present result for single quantum gates. In Section IV, we to efficiently solve some problems which are thought to be extend the analysis to multi-gate circuits with a test case of intractable for classical computers. Already several quantum two different implementations of a quantum full adder. Finally, algorithms are proposed including Shor’s prime factorization, we draw a conclusion in Section V. Grover’s search, Harrow-Hassidim-Lloyd’s linear system of II. BASICS OF QUANTUM COMPUTING AND SIMULATION equations which exhibit superior performance than their classi- FRAMEWORK cal counter-part and thus, pushing towards the goal of quantum supremacy [6]. The number of qubits is increasing with a 128- In this section, first, we introduce basic terminologies of qubit quantum computer is expected by 2019 [7]. More qubits quantum computing and then discuss the simulation frame- generally mean more computational power. work and underlying assumptions used in this paper. However, with existing technologies, qubit states are short A. Basic Terminologies lived. They tend to decohere due to different noise sources and State vector: Quantum computer comprises of qubits which Presented in GOMACTech 2019, March 25-28, 2019, Albuquerque, NM are the quantum analogue for classical bits. The state of a qubit at a time instance is given by the state vector j i. Generally, As Hamiltonian governs the evolution of qubit state, it is j i can be expressed as j i = a j0i + b j1i where, a and of paramount importance. At present, there are a number b are complex numbers such that jaj2 + jbj2 = 1. j0i and of different physical devices to realize quantum computers j1i are orthonormal set of basis vectors which span the state e.g. Ion trap [2], superconducting qubit [3], Silicon-based space of a physical system (also known as Hilbert space in [5], Nuclear Magnetic Resonance (NMR) based etc. quantum the context of quantum mechanics). Qubit can take a state computers. Each physical realization has its own definition of which is neither 0 norp 1, rather a superposition of both states. the Hamiltonian. For example, the Hamiltonian for a closed n j i = (j0i + j1i)= 2 is an example of such state that has spin coupled NMR system is given by [12]: 50% probability of being j0i and 50% probability of being X X J X D RF j1i. Fig. 1 graphically depicts the state vectors j0i, j1i and H = !kZk + Hj;k + Hj;k + H (2) their superposition state. k j;k j;k Density matrix: Besides state vector, state of a qubit can The first three terms describe the free precession of the spins be described with another entity known as density operator or in the ambient magnetic field, the magnetic dipole coupling density matrix which is defined by of the spins and the J coupling of the spins respectively X ρ ≡ p j i h j (a more complete description of these terms can be found i i i (1) RF i in [12]). However, the H term is particularly interesting where j ii is one of the numbers of states of the quantum from an external control standpoint. In presence of a large system and pi is the probability of that state. Although this radio-frequency (RF) field of a proper frequency, the unitary alternate description is mathematically equivalent to state vec- evolution of the system can be approximated as [12]: tor formulation, it provides convenient means of describing the H HRF (dynamic) evolution of a quantum system, especially systems exp(−i t) ≈ exp(−i t) (3) with noise. ~ ~ Time evolution and Hamiltonian: During quantum com- Equation (3) provides the insight that using external fields one putation, qubit states evolve with time as governed by can control the evolution (unitary transformation) of the quan- Schrodinger¨ equation. The evolution is a unitary transforma- tum system in an arbitrary way (although, different physical tion. If the state j i of a quantum system is evolved from realization imposes different constraints to this statement [12]. time t1 to t2, then the evolution is described by Schrodinger¨ However, each realization has at least a set of universal gates equation as follows: (unitary transformations) that can be implemented through d j i external control such as RF field in case of NMR quantum i = H j i computers). ~ dt H Open system and Master equation: Although Schrodinger¨ ) j (t2)i = exp(−i t) j (t1)i equation can describe the evolution of closed quantum sys- ~ y tems, it is not sufficient for open systems. An open system or, ρ(t2) = Uρ(t1)U is a quantum system that interacts with the environment. H [where, U = exp(−i t); t = t2 − t1] In reality, there are no perfectly isolated system and every ~ system is basically an open system. Interaction with environ- H is a Hermitian operator known as the system Hamiltonian ment portrays as noise in quantum computing and leads to which dictates the evolution of the closed system. U is the the quantum phenomenon named decoherence. An intuitive corresponding unitary matrix for Hamiltonian H. In gate- description of the problem related to decoherence can be based quantum computing regime, different gate operation is given using Schrodinger’s¨ cat analogy. The qubit can be at defined by different unitary matrices. For example, quantum a superposition of both j0i and j1i states simultaneously. NOT gate is defined by the following matrix (UNOT ) as it However, the environment works as an observer and according converts j i = j0i to j1i and vice versa. to the laws of quantum physics, this observation forces one of 0 1 1 0 1 0 the states to collapse leading to a classical state of either j0i U = and, = NOT 1 0 0 1 0 1 or j1i. This collapse of superposition state is a reason behind decoherence. |0 The dynamics of an open quantum system can be described |0> mainly with two methods. One approach is using the operator- sum method with Kraus operators [13]. An alternate approach Superposition of is to use the master equation, which can be written most basis states y generally in the Lindblad form [14] as: x |1> dρ i X y y y = − [H; ρ] + [2LjρL − L Ljρ − ρL Lj] (4) |1 dt j j j ~ j where H is the system Hamiltonian and, Lj are the Lindblad Fig.