Model Dynamics for Quantum Computing

Model Dynamics for Quantum Computing

Frank Tabakin Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA
July 6, 2019
Abstract

A model master equation suitable for quantum computing dynamics is presented. In an ideal quantum computer (QC), a system of qubits evolves in time unitarily and, by virtue of their entanglement, interfere quantum mechanically to solve otherwise intractable problems. In the real situation, a QC is subject to decoherence and attenuation effects due to interaction with an environment and with possible short-term random disturbances and gate deficiencies. The stability of a QC under such attacks is a key issue for the development of realistic devices. We assume that the influence of the environment can be incorporated by a master equation that includes unitary evolution with gates, supplemented by a Lindblad term. Lindblad operators of various types are explored; namely, steady, pulsed, gate friction, and measurement operators. In the master equation, we use the Lindblad term to describe short time intrusions by random Lindblad pulses. The phenomenological master equation is then extended to include a nonlinear Beretta term that describes the evolution of a closed system with increasing entropy. An external Bath environment is stipulated by a fixed temperature in two different ways. Here we explore the case of a simple one-qubit system in preparation for generalization to multi-qubit, qutrit and hybrid qubit-qutrit systems. This model master equation can be used to test the stability of memory and the efficacy of quantum gates. The properties of such hybrid master equations are explored, with emphasis on the role of thermal equilibrium and entropy constraints. Several significant properties of time-dependent qubit evolution are revealed by this simple study.

I Introduction

A quantum computer (QC) is a physical device that uses quantum interference to enhance the probability of getting an answer to an otherwise intractable problem Nielsen (); Preskill (). A quantum system’s ability to interfere depends on its entanglement and on maintenance of its coherent phase relations. In a real system, there are always environmental effects and also random disturbances that can cause the quantum system to lose its ability to display quantum interference. That process is called decoherence, as is discussed in an extensive literature Joos () on how a quantum system becomes classical, often rapidly, due to its interaction with an external environment. That process might also be viewed as a measuring device Zurek (); Hornberger (). A major concern in the development of a realistic quantum computer is to understand, control, and/or correct for detrimental environmental effects.

A general theory of how such “open systems” evolve in time is provided by the operator sum representation (OSR), which replaces the unitary evolution of a closed system by a more general form that accounts for the fact that the system under study (the quantum computer) is affected by an environment. That general form involves Kraus Kraus () operators and is often described as a mapping.

To gain insight, models for the environment and its interaction with the QC have been discussed Carmichael (). These studies use the Dyson series for the time evolution and reduce the dynamics to the QC subsystem using a variety of approximations. One approximation, truncating the Dyson series ( a Born approximation) is used since an exact solution is generally not available. An oft-used approximation is that in the system-environment interaction the environment restores itself rapidly to its initial condition, and therefore only the present situation of the environment is relevant. That is, one invokes a Markov approximation, which has the environment affecting the system, but the system’s effect on the environment vanishes rapidly. It is assumed that the system and environment are initially uncorrelated and are described as a product state.

The Markov approximation is not always applicable; that depends on the dynamics of the environment and its interaction with the system. It is physically possible that the system affects an environment that is able to partially preserve that influence and feed part of it back to the system. Indeed, there are important papers that indicate that the Markov approximation is in doubt Alicki (). Nevertheless, our initial approach is to adopt the Markov approximation, with plans to test its applicability.

A vast literature exists on deducing a general master equation for the time evolution of a subsystem’s density matrix. One method is to examine the above Dyson series methods, as described in Carmichael (). Another approach is to replace unitary evolution 111, by a Kraus subsystem form of evolution 222 with and .. Then a “generalized infinitesimal” expansion of the Kraus operators is used to deduce a differential equation for the density matrix, yielding a linear in density matrix master equation. The “generalized infinitesimal” is related to Ito calculus, which involves how to define variations for stochastic, or rapidly fluctuating functions. Several papers Adler (); Peres () discuss this procedure and ways to deduce a master equation for the subsystem’s density matrix, which prove to be of the form deduced earlier by Gorini, Kossakowski, Sudarshan and Lindblad  Lindblad1 (); Lindblad2 (); Lindblad3 (); Lindblad4 (), who used other general considerations. The Lindblad form can be deduced in general from requiring preservation of density matrix Hermiticity, unit trace, and positive-definite properties, without the restriction to time independent Lindblad operators or to particular initial states. The Lindblad form can be used to describe all environmental effects. However, we introduce special terms to isolate and to gain insight into specific effects, such as the Bath-system dynamics. Lindblad’s equation for the time evolution of a subsystem’s density matrix has many appealing features, as discussed later.

Profound papers by Beretta et al. Beretta1 (); Beretta2 (); Beretta3 () provide a general master equation based on novel concepts of non-equilibrium statistical mechanics as applied to quantum systems. The Beretta master equation for describing an open system has not been used much for QC perhaps because it is nonlinear and it alters entropy addition rules. An excellent rendition of the basic nature of the Beretta (and other nonlinear master equations ) is provided in papers by Korsch et al. Korsch ().

The Beretta et al. description includes definitions of entropy, work, and heat for non-equilibrium systems. Their resultant master equation has many important features, as illustrated in our study. Indeed, we incorporate their master equation for QC and extend it using a phenomenological viewpoint.

In section II, we discuss the basic idea of a density matrix from two traditional points of view. One view is to form a classical ensemble average over an ensemble of quantum systems. The second viewpoint is that a single quantum system is prepared in a state averaged over its production possibilities. After discussing general properties of the density matrix, we stress that the dynamics of the density matrix can be best visualized in terms of the time development of its spin polarization and spin correlation observables.

In section III, we describe how the density matrix evolves by unitary evolution driven by a time-dependent Hamiltonian that includes level splitting and ideal gate pulses. During a gate pulse, a bias pulse is used to impose temporary degeneracy to halt precession and avoid awkward phase accumulation. Then a model master equation is introduced in section IV along with an analysis of its several terms. These terms are: (1) Lindblad form used to include random noise and gate friction effects, (2) a Beretta term to describe a closed-system with increasing entropy, and (3) a Bath term for contact with an environment of specific temperature. We make a heuristic assumption that the Lindblad and Beretta forms are both useful, and we adopt a phenomenological or hybrid viewpoint. That viewpoint, which is not derived but postulated, is to use Lindblad terms to describe random, short-time intrusions on the QC, which could be caused by for example a passing particle. And we use the nonlinear Beretta term to describe the overall trend for a closed system to steadily increase in entropy. A Beretta description of Bath-system (open) dynamics is also included. The system is simultaneously driven towards equilibrium with an environment or bath of a specified temperature. On top of these major effects, we describe the time-evolution of quantum gates by Hamiltonian pulses. If the Lindblad and Bath terms are both set to zero, which is equivalent to removing the environment, and the closed-system Beretta term is omitted, then the master equation reduces necessarily to ordinary unitary evolution.

In section IV.3, we introduce the Lindblad master equation and discuss several aspects of its general features, and limitations. We then proceed to a similar analysis of the Beretta and Bath terms IV.4. Finally, in section V properties of the full master equation are examined and conclusions and future plans are stated in section VI.

The above assumptions provide a practical dynamic framework for examining not only the influence of an environment on the efficacy of a QC, but also the loss of reliability in the action of gates or the general loss of coherence. The master equation we design incorporates the main features of a density matrix; namely, Hermiticity, unit trace and positive definite character, while also including the evolution of a closed system and the effects of gates, noise and of an external bath.

Ii Density Operator

The density operator, also called a density matrix, is a operator in Hilbert space that represents an ensemble of quantum systems. As introduced by von Neumann and Landau Neumann (); Landau (); Fano (), the density operator, can be understood as a classical ensemble average over a collection of subsystems (the ensemble) which occur in a general state with a probability By a general state, we mean a state of the subsystem that is a general superposition of a complete orthonormal basis (such as eigenstates of a Hamiltonian). For the simple case of an ensemble of spin 1/2 particles, such a state called a qubit is specified by the spinor

(1)

where the computational basis and denote spin-up and spin-down states, respectively, and labels the Euler angles that specify a general direction in which the qubit is pointing. Indeed, the above state is an eigenstate of the operator where the components of are the Pauli operators ( see below). This description is readily generalized to multiparticle qubit states and also to systems that are doublets without being associated with the idea of physical spin.

The above general state is normalized but not necessarily orthogonal, The quantum rule for the expectation value of a general operator is and for an ensemble of separate quantum subsystems one can form the classical ensemble average for the Hermitian observable by taking

(2)

The ensemble average is then a simple classical average where is the probability that a particular state appears in the ensemble. Summing over all possible states of course yields . The above expression is a combination of a classical ensemble average with the quantum mechanical expectation value. It contains the idea that each member of the ensemble interferes only with itself quantum mechanically and that the ensemble involves a simple classical average over the probability distribution of the ensemble.

We now define the density operator by

(3)

Using closure 333The closure property, which is a statement that is a complete orthonormal basis, is , the ensemble average can now be expressed as a ratio of traces

(4)

which entails the properties

(5)

where denotes a complete orthonormal basis (such as the computational basis), and

(6)

which returns the original ensemble average expression.

ii.1 Properties of the Density Matrix

The definition is a general one, if we interpret as the label for the possible characteristics of a state. Several important general properties of a density operator follow from this definition. The density operator is:

  • Hermitian hence its eigenvalues are real;

  • has unit trace, hence the sum of its eigenvalues equals 1;

  • is positive definite, which means that all of its eigenvalues are greater or equal to zero. This, together with the fact that the density matrix has unit trace, ensures that each eigenvalue is between zero and one, and yet sum to 1.

  • for a pure state, every member of the ensemble has the same quantum state and only one appears and the density operator becomes . The state is normalized to one and hence for a pure state . Thus for a pure state one of the density matrix eigenvalues is 1, with all others zero.

  • for a general ensemble which has a mixture of possibilities as reflected in the probability distribution with the equal sign holding for pure states.

ii.1.1 Composite Systems and Partial Trace

For a composite system, such as an ensemble of quantum systems each of which is prepared with a probability distribution, the definition of a density matrix can be generalized to a product Hilbert space form involving systems of type A and B

(7)

where is the joint probability for finding the two systems with the attributes labelled by and For example, could designate the possible directions of one spin-1/2 system, while labels the possible spin directions of another spin 1/2 system, . One can always ask about the state of just system A or B by summing over or tracing out the other system. For example, the density matrix of system A is picked out of the general definition above by the following partial trace steps

(8)

Here we use the product space and we define the probability for finding system A in situation by

(9)

This is a standard way to get an individual probability from a joint probability.

It is easy to show that all of the other properties of a density matrix still hold true for a composite system case. It has unit trace, it is Hermitian with real eigenvalues and is positive definite.

ii.2 Comments about the Density Matrix

ii.2.1 Alternate Views of the Density Matrix

In the prior discussion, the view was taken that the density matrix implements a classical average over an ensemble of many quantum systems, each member of which interferes quantum mechanically only with itself. An alternate equally valid viewpoint is that a single quantum system is prepared, but the preparation of this single system is not pinned down. Instead all we know is that it is prepared in any one of the states labelled again by a generic state label with a probability . Despite the change in interpretation, or rather an application to a different situation, all of the properties and expressions presented for the ensemble average hold true; only the meaning of the probability is altered.

An important point concerning the density matrix is that the ensemble average (or the average expected result for a single system prepared as described in the previous paragraph) can be used to obtain these averages for all observables . Hence in a sense the density matrix describes a system and the system’s accessible observable quantities. It represents then an honest statement of what we can really know about a system. On the other hand, in Quantum Mechanics it is the wave function that tells all about a system. Clearly, since a density matrix is constructed as a weighted average over bilinear products of wave functions, the density matrix has less detailed information about a system than is contained in its wave function. Explicit examples of these general remarks will be given later.

To some authors the fact that the density matrix has less content than the system’s wave function, causes them to avoid use of the density matrix. Others find the density matrix description of accessible information as appealing. Indeed, S. Weinberg in recent papers Weinberg1 (); Weinberg2 () has advocated an interpretation of quantum mechanics based on using the density matrix rather than the state vector as a description of reality. The attribution of such deep physical meaning to the density operator was advocated earlier by Hatsopoulos and Gyftopoulos Hatsopoulos (), who inspired by a deep analysis by Park Park () on the nature of quantum states, adopted it as the key physical ansatz of their early theory of quantum thermodynamics, which in turn prompted Beretta to design the nonlinear master equation that we adopt below as part of our model master equation.

We now turn to discussing the basic features of the density matrix in preparation for describing its dynamic evolution by means of a model master equation.

ii.2.2 Classical Correlations and Entanglement

The density matrix for composite systems can take many forms depending on how the systems are prepared. For example, if distinct systems A & B are independently produced and observed independently, then the density matrix is of product form and the observables are also of product form For such an uncorrelated situation, the ensemble average factors

(10)

as is expected for two separate uncorrelated experiments. This can also be expressed as having the joint probability factor the usual probability rule for uncorrelated systems.

Another possibility for the two systems is that they are prepared in a coordinated manner, with each possible situation assigned a probability based on the correlated preparation technique. For example, consider two colliding beams, A & B, made up of particles with the same spin. Assume the particles are produced in matched pairs with common spin direction Also assume that the preparation of that pair in that shared direction is produced by design with a classical probability distribution Each pair has a density matrix since they are produced separately, but their spin directions are correlated classically. The density matrix for this situation is then

(11)

This is a “mixed state” which represents classically correlated preparation and hence any density matrix that can take on the above form can be reproduced by a setup using classically correlated preparations and does not represent the essence of Quantum Mechanics, e.g. an entangled state.

An entangled quantum state is described by a density matrix (or by its corresponding state vectors) that is not and can not be transformed into the two classical forms above; namely, cast into a product or a mixed form. For example, the two-qubit Bell state has a density matrix

(12)

that is not of simple product or mixed form. It is the prime example of an entangled state.

The basic idea of decoherence can be described by considering the above Bell state case with time dependent coefficients

(13)

If the off-diagonal terms vanish, by attenuation and/or via time averaging, then the above density matrix does reduce to the mixed or classical form,

(14)

which is an illustration of how decoherence leads to a classical state.

ii.3 Observables and the Density Matrix

Visualization of the density matrix and understanding its significance is greatly enhanced by defining associated real spin observables. In the simplest one-qubit case, the density matrix is a Hermitian positive definite matrix of unit trace. Thus it is fully stipulated by three real parameters, which are identified as the polarization vector also called the Bloch vector. One can deduce that only three parameters are needed for the one-qubit case from the following steps: (1) a general matrix with complex entries involves real numbers; (2) the Hermitian condition reduces the diagonal terms to 2 and the off-diagonal terms to 2, a net of 4 remaining real numbers; (3) the unit trace reduces the count by 1, so we have 4-3=3 parameters. These steps generalize to multi-qubit and to qutrit cases.

Operators or gates acting on a single qubit state are represented by 2 2 matrices. The dimension of the single qubit state vectors ( and ) is with The Pauli matrices provide an operator basis of all such matrices. The Pauli-spin matrices are:

(15)

These are all Hermitian traceless matrices . We use the labels to denote the directions The fourth Pauli matrix is simply the unit matrix. Any matrix can be constructed from these four Pauli matrices, which therefore are an operator basis, also called the computational basis operators. That construction applies to the density matrix at any time t and to the Hamiltonian and Lindblad operators

ii.3.1 Polarization

The general form of a one-qubit density matrix, using the 4 Hermitian Pauli matrices as an operator basis is:

where the spin operators are and the real polarization vector is The polarization, is a real vector, which follows from the Hermiticity of the density matrix and from the ensemble average relation

Thus specifying the polarization vector ( also called the Bloch vector) determines the density matrix and it is convenient to view the polarization as a function of time to gain insight into qubit dynamics.

The above expression clearly satisfies the density matrix conditions that The positive definite condition follows from determining that the two eigenvalues are where The unit trace condition becomes simply that the eigenvalues of sum to one

where is the unitary matrix that diagonalizes the density matrix at time t. The diagonal density matrix has real eigenvalues along the diagonal. The positive definite condition now asserts that each of these eigenvalues is greater or equal to zero and less than or equal to one: while summing to 1. For the one qubit case the above conditions mean that and since the polarization vector must have a length between zero and one.

Note that the density matrix, polarization vector and its eigenvalues in general depend on time. Indeed, the dynamics of a one-qubit system is best visualized by how the polarization or eigenvalues change in time.

The polarization operator is simply and we have the following relations for the value and time derivative of the polarization vector:

(18)

Much of what is presented here applies to multi-qubit and qutrit cases. The main difference for more qubits/qutrits is an increase in the number of polarization and spin correlation observables.

Several other quantities are used to monitor the changing state of a quantum system. Later energy, power, heat transfer and temperature concepts will be discussed. Next purity, fidelity, and entropy attributes will be examined.

ii.3.2 Purity

The purity is defined as It is called purity since for a pure state density matrix and but in general For a pure state, we see that implies that each eigenvalue satisfies so Since the eigenvalues sum to 1, a pure state has one eigenvalue equal to one, all others are zero. A mixed or impure state has which indicates that the nonzero eigenvalues are less than 1.

For a one-qubit system, the purity is simply related to the polarization vector

(19)

where is the length of the polarization vector . Thus a pure state has a polarization vector that is on the unit Bloch sphere, whereas an impure state’s polarization vector is inside the Bloch sphere. The purity ranges from a minimum of 0.5 to a maximum of 1. Later we will see how dissipation and entropy changes can bring the polarization inside the Bloch sphere and hence generate impurity.

ii.3.3 Fidelity

Fidelity measures the closeness of two states. In its simplest form, this quantity can be defined as For the special case that this yields which is clearly the magnitude of the overlap probability amplitude.

To align the quantum definition of fidelity with classical probability theory, a more general definition is invoked; namely,

(20)

When and commute, they can both be diagonalized by the same unitary matrix, but with different eigenvalues. In that limit, we have and

(21)

which is the classical limit result.

We will use fidelity to monitor the efficacy or stability of any QC process, where is taken as the exact result and is the result including decoherence, gate friction, and dissipation effects.

ii.3.4 Entropy

The Von Neumann Neumann () entropy at time t is defined by

(22)

The Hermitian density matrix can be diagonalized by a unitary matrix at time t,

where is diagonal matrix of the eigenvalues. Then

(23)

With a base 2 logarithm, the maximum entropy for one qubit is which occurs when the two eigenvalues are all equal to That is the most chaotic, or least information situation. The minimum entropy of zero obtains when one eigenvalue is one, all others being zero; that is the most organized, maximum information situation. For one qubit, zero entropy places the polarization vector on the Bloch sphere, where the length of the polarization vector is one. If the polarization vector moves inside the Bloch sphere, entropy increases. For qubits entropy ranges between zero and .

For later use, consider the time derivative of the entropy

Since the second RHS term above vanishes. Note that the above result is derived assuming that for all eigenvalues which is ambiguous for zero eigenvalues. This is no doubt related to divergences that could arise when say and is nonzero. We will confront this issue later.

Note that the eigenvalues, purity, fidelity and entropy all depend on the length of the polarization vector .

Iii First Steps towards a Master Equation Model– Unitary Evolution, Gates and Pulses

The master equation for the time evolution of the system’s density matrix is now presented. We are interested in developing a simple model that incorporates the main features of the qubit dynamics for a quantum computer. These main features include seeing how the dynamics evolve under the action of gates and the role of both closed system dynamics and of open system decoherence, dissipation and the system’s approach to equilibrium. From the density matrix we can determine a variety of observables, such as the polarization vector, the power and heat rates, the purity, fidelity, and entropy all as a function of time.

iii.1 Unitary evolution

We start with the observation that the density matrix for a closed system is driven by a Hamiltonian that can be explicitly time dependent, as where the unitary operator is For infinitesimal time increments this yields the unitary evolution or commutator term:

(25)

This term specifies the reversible motion of a closed system. To include dissipation, an additional operator will be added which describes an irreversible open system.

iii.2 Hamiltonian

Our Hamiltonian is an Hermitian operator in spin space; for one qubit it is a matrix. It consists of a time independent plus a time dependent part For , a typical Hamiltonian is

(26)

which describes a 2 level system with eigenvalues for state and for state see Fig. 1.

Figure 1: The qubit levels with splitting With our conventions the operator raises the qubit to the polarization-down state while lowers the qubit to the polarization-up ground state .

The polarization vector for this case precesses about the direction with the Larmor angular frequency This follows from the unitary evolution term

where which is a Larmor precession of the polarization vector about the direction The polarization vector then has a fixed value of and the x and y components vary as

(28)

The above is equivalent to with

(29)

This form will be extended to dissipative cases later.

Thus the level splitting produces a precessing polarization with a fixed z-axis value and circular motion in the x-y plane ( see Fig. 2).

Figure 2: Polarization vector precession (no gates and no dissipation): (a) fixed and oscillating components versus time, (b) the two (fixed) eigenvalues of (c) the fixed entropy, and (d) the fixed energy (power and heat rate are zero). The Bloch sphere (e) with solid vector indicating the initial location of the polarization (which originates from the center of the Bloch sphere) while the subsequent motion follows the thick path as also shown by the dashed polarization vectors at subsequent times. The dots indicate equal time interval locations of the polarization vector. The precession is also projected to the x-y plane. The initial density matrix and level parameters are in Table 1.

The basic Hamiltonian is selected to be time independent. The initial density matrix and level splitting parameters used in our examples 444All of the numerical examples in this paper were generated by Mathematica codes based on the QDENSITY/QCWAVE packages qdensity (). are listed in Table 1. Energy is in eV, frequency in GHz and time is in nanoseconds (nsec).

Name Value
0.5
0.0
0.8
0.943
Initial Purity 0.945
Initial Entropy 0.186
Initial Temperature 0.93 mK
Larmor frequency 0.2675 GHz
Larmor Period 23.5 nsec
Level split 0.1761 eV
Table 1: Initial Density Matrix & Level Parameters

Next we add a time dependence in the form of Hamiltonian pulses that produce quantum gates.

iii.2.1 One-qubit ideal gates

For our QC application, the Hamiltonian is used to incorporate two effects. The first is the level splitting, Eq. 26. Here denotes the Larmor angular frequency associated with the level splitting which sets the Larmor time scale for the system. The term is used to include quantum gates which are Hermitian matrices. For example, a single qubit NOT gate is The NOT acts as: This basic gate is simply a spinor rotation about the axis by radians. Clearly, two NOTs return to the original state.

A gate operator is introduced as a Hamiltonian generator

(30)

where is a gate pulse that is centered at time with a width The pulse has inverse time units. Thus the pulse essentially starts at and ends at we typically take this pulse to be of Gaussian form,

(31)

We call the gate generator 555 The unitary operator associated with this gate generator is: . since it generates the effect of a specific gate. The pulse function is designed to generate a suitable rotation over an interval to Since we want to have a smooth pulse, we take these pulses to be of either Gaussian or soft square shape. The soft square shape is defined by

where is fixed by the condition.

The NOT gate pulse represents a series of infinitesimal rotations about the x-axis and in order to give the correct NOT gate effect, we need to normalize the pulse by The same form can be applied to a one qubit Hadamard

(32)

which is a spinor rotation about the axis by radians.

iii.2.2 Bias gates

Application of such a gate pulse does not carry out our objective of achieving a NOT gate, unless we do something to remove the level splitting at least during the action of the pulse. This corresponds to a temporary stoppage of precession. We therefore, introduce a bias pulse which is designed to make the levels degenerate during the gate pulse. The strength of the bias is adjusted by some type of non-intrusive monitoring, or by fore-knowledge of the fixed level splitting, to temporarily establish level degeneracy. During the action of the gate, the levels have to be completely degenerate, otherwise disruptive phases accumulate. Therefore, we use a soft square bias pulse that straddles the time interval of the gate pulse. The soft square bias pulse shape is defined by: which is preferred over a square pulse since it has finite derivatives and thus yields smooth variations of power as shown later. The width of the above bias pulse is and is the thickness of the edges. To be sure that no precession occurs during a gate the time values used in the bias pulse and are taken to be slightly larger and slightly smaller than the gate pulse values and

The bias pulse is added to the Hamiltonian to create a temporary degeneracy as

(33)

where the bias normalization is Note is unitless, whereas has 1/time units.

Combining these terms we have for a single pulse, with gate and bias

Here we see that the bias turns off precession and the gate term generates the action of a gate Without a bias pulse to produce level degeneracy, awkward phases accumulate that are detrimental to clean-acting gates. Aside from intervals when the gate and the bias pulse act, the polarization vector precesses at the Larmor frequency, which is zero for degenerate levels. The bias pulse is simply an action to stop the precession, then the gate pulse rotates the qubit, and subsequently precession is restored once the bias is removed. That process is equivalent to stopping a spinning top, rotate it, and then get it spinning again, which requires some work. As discussed later the power supplied to the system during a gate pulse is determined by

The derivative of the Hamiltonian divides into a gate plus a bias term.

iii.2.3 Gate and Bias Cases

In Figures 3-4, the one-qubit polarization vector motion for a NOT and a Hadamard gate are shown with no dissipation () and with a bias pulse acting during the gate pulse. The detailed case shows that during the NOT pulse one gets the expected change of The power supplied to the system during the NOT gate is also displayed separately for the gate power and the bias power. These are explained by the bias power and the gate power where the x-polarization is fixed during the NOT gate, but the z-polarization flips. The values of the polarization from the time when the gate pulse starts to its end at explain the shapes seen in Fig 3.

During the Hadamard pulse one gets the expected change of The power supplied to the system during the Hadamard gate is also displayed separately for the gate power and the bias power. These are explained by the bias power and the gate power where the y-polarization flips during the Hadamard gate, and the z and x polarization interchange. The values of the polarization during the pulse explain the shapes seen in Fig 4.

The gate pulses can produce net work done on the system. No heat transfer occurs by way of the gate or bias, that exchange arises later from dissipation. After the gate pulses are complete, the precession continues about the axis.

Another case of a Hadamard gate is shown in Fig. 5. In this case, the Hamiltonian is smoothly rotated from to during the Hadamard gate pulse. This Hamiltonian rotation, which is equivalent to rotating a level splitting magnetic field from the z to x direction, is accomplished by setting:

(34)

Here is a smooth step function of width As a result the precession which started around the continues about the axis after the Hadamard gate pulse as shown in Fig. 5.

Figure 3: Polarization vector trajectory for unitary Not Gate with level splitting and bias. Dissipation is off (a) changes in polarization with time, during Not Gate pulse ; (b) the two (fixed) eigenvalues of (c) the fixed entropy ; (d) the energy versus time ; (e)Power by gate (G) and bias (B) (heat rate is zero). here is negative during the gate ; and (f) precession about positive is moved to axis by NOT gate. The dots indicate equal time interval locations of the polarization vector.
Figure 4: Polarization vector trajectory for a unitary Hadamard gate with level splitting and bias. Dissipation is off (a) Polarization versus time; (b) Polarization evolution during Hadamard gate pulse(G) when gate starts   nsec, (Px,Py,Pz)=(-0.402,-0.301,0.798) ; (c) the two (fixed) eigenvalues of (d) the energy versus time ; (e) Power by gate (G) and bias (B) (heat rate is zero) ; and (f) Polarization vector trajectory, precession about positive continues after polarization is moved as shown by Hadamard gate.
Figure 5: Polarization vector trajectory for unitary Hadamard Gate plus bias and with off. The precession axis is changed from the to the axis during the gate pulse. (a) changes in polarization during Hadamard gate pulse; (b) the two (fixed) eigenvalues of (c) the fixed entropy ; (d) the energy versus time changes due to gate, bias and Hamiltonian axis rotation; (e) Power by gate , bias and Hamiltonian rotation (heat rate is zero) ; (f) precession starts about axis and continues about axis after polarization is moved as shown by Hadamard gate. The dots indicate equal time interval locations of the polarization vector.

iii.2.4 Gate pulses and instantaneous gates

To fully replicate the results obtained when a set of instantaneous gates act, as in a QC algorithm, it is necessary to invoke additional steps. One possible step is to apply a bias pulse over the full set of gate pulses, thereby making the qubits degenerate during a QC action, including final measurements. Another way, which we prefer, is to let the precession continue between gate pulses, which means that each gate acts with an associated bias pulse, as illustrated earlier. Then one needs to design the gate pulses and associated measurements to act at appropriate times to replicate the standard description of instantaneous gates. For example, we define a delay time as an integer multiple of the Larmor period The first gate starts at a time The first pulse ends at a time The next gate starts at a time and ends at a time This setup repeats for gates and yields the final time that we use to define the completion of the QC process as At the time the action of the gates is complete and the corresponding density matrix is the same as the instantaneous, static gate result where is a product of the gate operators. The general result for the final time is

For example, consider a three gate case for one qubit which is a three gate Hadamard-Not-Hadamard sequence. This case is illustrated in Fig. 6. At the first stage before the gate acts, the polarization vector precesses about the z-axis, then the Hadamard acts at time and the polarization path moves rapidly to the second lower precession circle at time After a few precessions, the Not gate brings the path to the negative z region at time Finally, the final Hadamard lifts the path back up to the original precession cone, but with a phase change. The final result at time is obtained by the transformation which is, as it should be, equivalent to the action of a single gate. The projected version also shown in Fig. 7 displays this process and the finite time dynamic gate actions.

This process can be implemented for any set of gate pulses and can be generalized to multi-qubit/qutrit cases. Thus the pulsed gate approach can replicate the standard instantaneous static gate description by carefully designing the timing of the gates on the Larmor precession time grid. This requires examining or measuring the gates at the selected time If the Larmor period varies in time, this procedure can be generalized.

Figure 6: Polarization for Hadamard-Not-Hadamard pulse gates. (a) Polarization vectors, (b) Eigenvalues, (c) Entropy, (d) Energy of first pulse, (e) Energy of second pulse, all versus time. Then in (f) Trajectories and initial and final polarization vectors, along with the three rapid gate pulses. The initial polarization vector (solid green arrow ) and final polarization vector (dashed purple arrow ) are seen to give the expected gate result.
Figure 7: Polarization trajectories for Hadamard-Not-Hadamard pulse gates projected onto the y-z plane. The first Hadamard downward pulse is seen with the next Not downward pulse, and then the final upward Hadamard pulse. In this example, the initial polarization vector (solid green arrow ) and final polarization vector (dashed purple arrow ) are seen projected onto the y-z plane.

For a sequence of gates

where is the ith gate acting at the time centered at This can generate a chain of gates.

We conclude that one can replicate the action of instantaneous static gates, which is central to the usual description of QC algorithms, by including a bias pulse during the gate action, and by applying the gates on the Larmor period time-grid.

iii.3 Schrödinger, Heisenberg, Dirac ( Interaction) and Rotating Frame Pictures

In our treatment, we use the Schrödinger picture for the density matrix, so that all aspects of the dynamics are described by the density matrix through its polarization and spin correlation observables. Other choices are to either use the Heisenberg picture, where the time development is incorporated into the Hermitian operators, or use the Dirac or Interaction picture, wherein the operators evolve in time with the “free” Hamiltonian Then the Dirac picture density matrix evolves as:

(36)

where the tilde denotes interaction picture operators. For the choice of going to the Dirac picture is simply going to a frame rotating about the z-axis in which frame the Larmor precession vanishes. That is called the rotating frame. Since we include gates into and hence , the gate bias pulse that we introduce in the Schrödinger picture, corresponds to a rotating frame that stops rotating during the action of a gate.

There are advantages offered by each of these choices. We stick with the Schrödinger description because it most clearly reveals the full dynamics by viewing the time evolution of the spin observables.

Iv The Master Equation Model

The master equation for the time evolution of the system’s density matrix is now presented. We seek a simple model that incorporates the main features of qubit dynamics for a quantum computer. These main features include seeing how the dynamics evolve under the action of gates and the role of both closed system dynamics and of open system decoherence, dissipation and the system’s approach to equilibrium. From the density matrix we can determine a variety of observables, such as the polarization vector, the power and heat rates, the purity, fidelity, and entropy all as a function of time.

iv.1 Definition of the Model Master Equation

To the unitary evolution, we now we add a term which is required to be Hermitian and traceless so that the density matrix maintains its hermiticity and trace one properties. In addition, has to keep positive definite. To identify explicit physical effects, we separate into three terms:

(37)

the operator involves a base e logarithm to assure that a Gibbs density matrix is obtained in equilibrium (see later). The QC entropy is defined with a base 2 operator with entropy equal to The conversion factor is and with The level splitting, gates and bias pulses are included in The state dependent, and hence time dependent, functions will be defined later.

When we discuss equilibrium, a form that combines the Beretta and Bath terms is used:

(38)

with and