Principles and Demonstrations of Quantum Information Processing by NMR Spectroscopy^{∗}

May 8, 1999

Abstract

This paper surveys our recent research on quantum information processing by nuclear magnetic resonance (NMR) spectroscopy. We begin with a brief introduction to the product operator formalism, on which the theory of NMR spectroscopy is based, and use it throughout the rest of the paper to show how it provides an concise framework within which to analyze quantum computations and decoherence. The implementation of quantum algorithms by NMR depends upon the availability of special kinds of mixed states, called pseudo-pure states, and we consider a number of different methods for preparing pseudopure states, along with what is known about how they scale with the number of spins. The quantum-mechanical nature of processes involving such macroscopic pseudo-pure states also is a matter of debate, and

∗Portions of this survey were presented at the AeroSense Workshop on Photonic Quantum Computing II, held in Orlando, Florida on April 16, 1998, at the Dagstuhl Seminar on Quantum Algorithms, held in Schloss Dagstuhl, Germany on May 10 – 15, 1998, and at the Workshop on Quantum Information, Decoherence and Chaos, held on Heron Island, Australia September 21 – 25, 1998; this paper is an updated and extended version of one published in the proceedings of the AeroSense meeting, available as vol. 3385 from the International Society for Optical Engineering, 1000 20th St., Bellingham, WA 98225, USA.

we attempt to make this debate more concrete by presenting the results of NMR experiments which validate Hardy’s paradox, subject to certain assumptions that we explicitly state. Finally, a detailed product operator description is given of recent NMR experiments which demonstrate the principles behind a three-bit quantum error correcting code.

1 INTRODUCTION

An ensemble quantum computer consists of a massive collection of independent and identical quantum computers, which all operate on the same coherent superposition in lock-step, and for which one can determine the sum of their one-bit observables over the ensemble in a single measurement. This enables it to use classical ensemble parallelism to estimate the expectation values of the observables, rather than random eigenvalues thereof [12]. Although we do not expect any asymptotic performance gains with an ensemble of fixed size, an ensemble quantum computer retains all the essential features of a quantum computer, including entanglement and interference, and hence can solve the same problems in polynomial time. The further ability to estimate expectation values can potentially reduce the amount of time that would be needed on an equivalent quantum computer by a very large, albeit constant factor, for example when using Grover’s algorithm to solve search problems [5, 19].

The main reason an ensemble quantum computer is interesting, however, is that it can be implemented with surprising ease, at least on a limited scale, by NMR spectroscopy on ordinary liquids at room temperature and pressure. This implementation takes advantage of several very favorable features of NMR spectroscopy on the spin = 1/2 nuclei of molecules in a liquid, in particular their seconds-long decoherence times. The implementation also depends on the availability of special kinds of mixed states, whose behavior under coherent Hamiltonians is identical to that of true pure states [12, 18]. The general methods that are presently available for preparing such pseudo-pure states from the equilibrium state of the spins in a magnetic field result in a rapid loss of observable magnetization with increasing number of spins. This in turn restricts such implementations to ca. 8 – 10 spins in the foreseeable future. Although this is certainly a severe restriction, NMR has now provided direct experimental demonstrations of many of the basic elements of quantum information processing [10, 11, 14], including simple quantum algorithms [8, 9, 22, 23], teleportation [30], and error correcting codes [13].

This paper will survey our recent research on quantum information processing by NMR spectroscopy. We begin with a brief description of the mathematical framework within which NMR experiments are widely analyzed, known as the product operator formalism, and show how it applies to quantum computing (cf. [34]). Next, we give an overview of the basic ideas behind ensemble quantum computing by NMR spectroscopy, with emphasis on pseudo-pure state preparation and scaling. We then consider some ways in which quantum correlations can be manifest even in weakly polarized spin ensembles, and illustrate this with the results of NMR experiments which (with certain assumptions) validate Hardy’s paradox [21]. Finally, the utility of NMR and its associated product operator formalism as a means of studying decoherence will be demonstrated by an analysis of our recent experiments with a three-bit quantum error correcting code [13]. The reader is assumed throughout to be familiar with the basic notions of quantum information processing, as presented in e.g. Refs. [31, 37, 40]. A more introductory account of our work on ensemble quantum computing by NMR spectroscopy may be found in Ref. [14].

2 THE PRODUCT OPERATOR FORMAL-ISM

The basic observables in the NMR spectroscopy are the x- and y-components of the spin angular momentum (traditionally in units of h̄). These give rise to absorptive and dispersive peaks, respectively, centered on the resonance frequencies of the spin in the applied static magnetic field [33]. The operators for the angular momentum components of a spin are denoted by I_{x}, I_{y} and I_{z}, which for the spin = 1/2 nuclei dealt with in this paper are typically represented by one half the Pauli matrices. Although products of the angular momentum components of different spins cannot be observed directly, they evolve naturally into single components, and hence it is also natural to define operators for such products. This can be done using the mixed product formula between the operator composition product and tensor products:

(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)  (1)

Thus we have, for example, the following operator for the product of the x component of the first spin with the y component of a second:

⟨ϕ^{1}|I_{x}^{1}|ϕ^{1}⟩⟨ψ^{2}|I_{y}^{2}|ψ^{2}⟩ = ⟨ϕ|I_{x}|ϕ⟩ ⊗ ⟨ψ|I_{y}|ψ⟩
= (⟨ϕ| ⊗ ⟨ψ|)(I_{x} ⊗ I_{y})(|ϕ⟩ ⊗ |ψ⟩)
= ⟨ϕ^{1}ψ^{2}|(I_{x} ⊗ 1)(1 ⊗ I_{y})|ϕ^{1}ψ^{2}⟩
= ⟨ϕ^{1}ψ^{2}|I_{x}^{1}I_{y}^{2}|ϕ^{1}ψ^{2}⟩

(2)

Operators like I_{x}^{1}I_{y}^{2} are called product operators [4, 17, 34, 36, 38]. The mixed product formula, together with the multiplication rules inherited from the Pauli matrix algebra, gives us a well-defined multiplication between any pair of product operators. Henceforth, we shall identify the identity operator 1 with the scalar identity 1.

Product operators enable us to represent not only the observables of NMR, but also the spin states and their transformations within a single algebraic system. To explain this, we first note that the state of the spins in an NMR sample at room temperature is necessarily highly mixed, and so must be described by a density operator [3]. We shall also use density operators to describe pure states. For example, the density operator of two spins antiparallel along the z-axis is

|01⟩⟨01| = (|0⟩⟨0|) ⊗ (|1⟩⟨1|) = \frac{1}{2}(1 + 2I_{z}^{1})\frac{1}{2}(1 − 2I_{z}^{2}) ≡ E_{+}^{1}E_{−}^{2} . (3)

It is easily verified that the operators E_{±}^{k} = \frac{1}{2}(1 ± 2I_{z}^{k}) of the individual spins are idempotent, meaning they are equal to their squares, and that they are mutually annihilating, i.e. E_{+}^{k}E_{−}^{k} = 0 = E_{−}^{k}E_{+}^{k}. A general pure state corresponds to a primitive idempotent, meaning it cannot be expressed as a sum of two nonzero idempotents, and the pure state is uncorrelated if it can be factorized into one-spin idempotents (as above). A general mixed state is given by a sum of (not necessarily factorizable) primitive idempotents with nonnegative coefficients p_{i} summing to one, which are the probabilities of the corresponding pure states in the thermodynamic ensemble:

ρ ≡ \overline{|ψ⟩⟨ψ|} = ∑_{i} p_{i} |ψ_{i}⟩⟨ψ_{i}|  (4)

This representation of quantum states by density operators makes it clear that, if U is a unitary transformation which acts on state vectors by leftmultiplication |ψ⟩ → U|ψ⟩, then it acts on the corresponding density operator by conjugation, i.e.

ρ = \overline{|ψ⟩⟨ψ|} ⟶ \overline{(U|ψ⟩)(U|ψ⟩)^{∼}} = U\overline{|ψ⟩⟨ψ|}Ũ = Uρ Ũ (5)

(where Ũ ≡ U^{∼} denotes the Hermitian conjugate). It is also easily seen that the ensemble-average expectation value of any observable A is given by the trace of its product with the density operator:

\overline{⟨ψ|A|ψ⟩} = Tr(\overline{⟨ψ|A|ψ⟩}) = Tr(A\overline{|ψ⟩⟨ψ|}) ≡ Tr(Aρ)  (6)

This in turn is just 2^{N} times the (real) scalar part ⟨Aρ⟩ ≡ 2^{−N} Tr(Aρ) of the product.^{1} For example, using the relations 2⟨E_{±}^{k}⟩ = 1 and 2E_{±}^{k}I_{z}^{k} = ±E_{±}^{k}, we can show that the z-component of the total angular momentum depends on the polarizations p and q of two uncorrelated spins as

\begin{aligned}& 4 ⟨((1 − p)E_{+}^{1} + p E_{−}^{1})((1 − q)E_{+}^{2} + q E_{−}^{2})(I_{z}^{1} + I_{z}^{2})⟩ \\ = & 4 ⟨(1 − p)(1 − q)E_{+}^{1}E_{+}^{2} − pq E_{−}^{1}E_{−}^{2}⟩ \\ = & (1 − p)(1 − q) − pq = 1 − p − q . \end{aligned} (7)

Turning now to the transformations, we first note that NMR spectroscopists typically use an interaction representation, wherein all the transformations and states are referred to a right-handed coordinate frame which

1The factor of 2^{−N} in the scalar part is not as unnatural as it may seem, since as we shall see it also appears in the partition function.

co-rotates at the receiver frequency about the axis of precession z [33]. The unitary transformation that corresponds to a right-hand rotation of the k-th spin by an angle θ about the x axis in this frame is given by

e^{−ıθI_{x}^{k}} = 1 − ı (\frac{θ}{2}) 2I_{x}^{k} − \frac{1}{2} (\frac{θ}{2})^{2} + \frac{ı}{6} (\frac{θ}{2})^{3} 2I_{x}^{k} + ⋯
= cos (\frac{θ}{2}) − ı sin (\frac{θ}{2}) 2I_{x}^{1} ,

(8)

where we have used (2I_{x}^{k})(2I_{x}^{k}) = 1. In NMR, this operation is implemented by an RF (radio-frequency) pulse whose frequency range spans the resonances of the k-th spin. We shall also encounter correlated rotations, in particular that induced by the bilinear interaction known in NMR as scalar coupling,

e^{−ı t H_{J}} = e^{−ı t 2πJ^{12}I_{z}^{1}I_{z}^{2}} = cos (\frac{π}{2}J^{12}t) − ı sin (\frac{π}{2}J^{12}t) 4I_{z}^{1}I_{z}^{2} , (9)

where H_{J} is the (weak) coupling Hamiltonian and J^{12} the coupling constant in Hz. As we shall see momentarily, this interaction permits us to implement general quantum logic gates by NMR.

In quantum computing, the NOT operation on the k-th spin simply rotates it by π; according to the above formula:

e^{−ıπI_{x}^{k}}E_{+}^{k}e^{ıπI_{x}^{k}} = (−2ıI_{x}^{k})E_{+}^{k}(2ıI_{x}^{k}) = \frac{1}{2}(1 + 8I_{x}^{k}I_{z}^{k}I_{x}^{k})
= \frac{1}{2}(1 − 8I_{x}^{k}I_{x}^{k}I_{z}^{k}) = \frac{1}{2}(1 − 2I_{z}^{k}) = E_{−}^{k}

(10)

The c-NOT (controlled-NOT) operation is a π rotation of e.g. the first spin conditional on the polarization of a second. Using the relation E_{±}^{k}E_{∓}^{k} = 0, we can easily show that

(−2ıI_{x}^{1}E_{−}^{2} + E_{+}^{2})(E_{ϵ}^{1}E_{±}^{2})(2ıI_{x}^{1}E_{−}^{2} + E_{+}^{2}) = E_{±ϵ}^{1}E_{±}^{2}  (11)

(ϵ ∈ {±}). The phase factor ı multiplying I_{x}^{1} complicates the action of the c-NOT on a superposition, but can be eliminated by a conditional phase shift. Using E_{−}^{k} + E_{+}^{k} = 1, this phase-corrected c-NOT gate is seen to be S^{1|2} ≡

\begin{aligned}2I_{x}^{1}E_{−}^{2} + E_{+}^{2} & = & (−ıE_{−}^{2} + E_{+}^{2})(2ıI_{x}^{1}E_{−}^{2} + E_{+}^{2}) \\ & = & (1 + (e^{−ı \frac{π}{2}} − 1)E_{−}^{2}) (1 + (e^{ıπI_{x}^{1}} − 1)E_{−}^{2}) \\ & = & e^{−ı \frac{π}{2} E_{−}^{2}}e^{ıπI_{x}^{1}E_{−}^{2}} = e^{−ı \frac{π}{2} (1−2I_{x}^{1})E_{−}^{2}} , \end{aligned}  (12)

and hence the idempotents E_{±}^{k} also give us an algebraic description of the c-NOT gate, in addition to the density operators of pure states. It is wellknown that single spin rotations, together with the c-NOT, are sufficient to implement any quantum logic gate [2].

It is also well-known how these basic operations can be implemented by NMR [14]. The individual spins can be rotated by selective radio-frequency pulses, as mentioned above. Conditional rotations, and in particular the c-NOT, can be implemented by combining selective rotations with scalar coupling. Noting that

e^{ı \frac{π}{2} I_{x}^{1}}e^{−ı \frac{π}{2} I_{y}^{1}} = e^{ı \frac{π}{2} I_{z}^{1}}e^{ı \frac{π}{2} I_{x}^{1}} ,  (13)

we obtain:

S^{1|2} = e^{−ı \frac{π}{2} E_{−}^{2}}e^{ıπI_{x}^{1}E_{−}^{2}}
= e^{−ı \frac{π}{4} (1−2I_{z}^{2})}e^{ı \frac{π}{2} I_{x}^{1}(1−2I_{z}^{2})}
= e^{−ı \frac{π}{4}} e^{ı \frac{π}{2} I_{z}^{2}}e^{ı \frac{π}{2} I_{x}^{1}}e^{−ıπI_{x}^{1}I_{z}^{2}}
= e^{−ı \frac{π}{4}} e^{ı \frac{π}{2} I_{z}^{2}}e^{ı \frac{π}{2} I_{x}^{1}}e^{−ı \frac{π}{2} I_{y}^{1}}e^{−ıπI_{z}^{1}I_{z}^{2}}e^{ı \frac{π}{2} I_{y}^{1}}
= e^{−ı \frac{π}{4}} e^{ı \frac{π}{2} I_{z}^{2}}e^{ı \frac{π}{2} I_{z}^{1}}e^{ı \frac{π}{2} I_{x}^{1}}e^{−ıπI_{z}^{1}I_{z}^{2}}e^{ı \frac{π}{2} I_{y}^{1}}

(14)

Since the overall phase of the transformation has no effect on the density operator, it follows that the c-NOT S^{1|2} can be implemented in NMR by applying the following sequence of effective Hamiltonians:

[−\frac{π}{2}I_{y}^{1}] → [πI_{z}^{1}I_{z}^{2}] → [−\frac{π}{2}I_{x}^{1}] → [−\frac{π}{2}(I_{z}^{1} + I_{z}^{2})]  (15)

In practice, the effective Hamiltonian [πI_{z}^{1}I_{z}^{2}] is obtained by applying a πpulse to both spins in the middle and at the end of a 1/(2J^{12}) evolution period, to refocus the Zeeman evolution [14]. The [−\frac{π}{2}I_{y}^{1}] and [−\frac{π}{2}I_{x}^{1}] Hamiltonians are implemented by RF pulses as above, while the [−\frac{π}{2}(I_{z}^{1} + I_{z}^{2})] transformation can most easily be implemented by letting one spin evolve while refocusing the other, then vice versa, and finally realigning the transmitter’s phase with the spins’.

In closing this section, we note that the algebra of a single spin is isomorphic to the Clifford algebra of space [34]. This is naturally contained within the Dirac matrix algebra, which in turn is isomorphic to the Clifford algebra of space-time. A relativistic multiparticle version of the latter is given by the Clifford algebra of a direct sum of multiple copies of space-time, and contains an algebra isomorphic to the product operator algebra [15, 16]. Thus, the product operator algebra is not just a specialized tool for NMR spectroscopists, but plays a fundamental role in physics.

3 PSEUDO-PURE STATE PREPARATION AND SCALING

In liquids, the dipole-dipole interaction among the spins is averaged to zero by the rapid rotational diffusion of the molecules. This effectively makes each molecule into an independent quantum computer. It also enables the density operator to be factorized into a product of density operators for the individual molecules,

ϱ = ϱ^{1} ⋯ ϱ^{M} = ρ^{1} ⊗ ⋯ ⊗ ρ^{M} ,  (16)

where ϱ^{k} ≡ 1 ⊗ ⋯ ⊗ ρ^{k} ⊗ ⋯ ⊗ 1. In a pure liquid (or if we are looking at just one component of a solution), all the molecules are equivalent so that all these density operators are the same. It follows that we can work with a reduced density operator ρ ≡ ρ^{1} = ⋯ = ρ^{M} of dimension 2^{N}, rather than 2^{MN}.

The equilibrium density operator ρ_{eq} of an N-spin system in a magnetic field is given by the Boltzman operator determined by its Hamiltonian, exp(−H/k_{B}T), divided by the partition function Z_{eq} = Tr(exp(−H/k_{B}T)). At “high temperatures” (which, given the gyromagnetic ratios of nuclear spins, means more than a few mK), H is very small compared to k_{B}T, so that a linear approximation is quite accurate:

ρ_{eq} ≈ (1 − H/k_{B}T)/Tr(1 − H/k_{B}T) = (1 − H/k_{B}T)/2^{N}  (17)

The identity component 1/2^{N} gives rise to no net magnetization, since it represents a state in which each spin is up as often as it is down. The Hamiltonian H, on the other hand, is well-approximated by the dominant Zeeman precession term H_{Z}, which can be written as

H_{Z} ≡ − h̄ B_{z}(γ^{1}I_{z}^{1} + ⋯ + γ^{N}I_{z}^{N})  (18)

where γ^{k} is the gyromagnetic ratio of the k-th spin, and B_{z} is the z-component of the magnetic field. In practice, NMR spectroscopists usually drop the unobservable identity component from the density operator. Moreover, in homonuclear systems they also leave out both the constant factor h̄B_{z}γ^{k} as well as the partition function 2^{−N} — though we shall carefully avoid doing the latter! In these terms, the equilibrium density operator of a two-spin system, and its matrix representation in the standard I_{z}-eigenbasis, is

ρ̂_{eq} = \frac{1}{4}(I_{z}^{1} + I_{z}^{2}) = \frac{1}{4}(|00⟩⟨00| − |11⟩⟨11|)

↔ \frac{1}{4} Diag(1, 0, 0, −1) ≡ \frac{1}{4} [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ] , (19)

where the “hat” on ρ_{eq} signifies its traceless part.

In contrast, the density operator of two spins in their pseudo-pure ground state is

ρ̂_{00} ≡ ±\frac{1}{6}(I_{z}^{1} + I_{z}^{2} + 2I_{z}^{1}I_{z}^{2}) = ± \frac{1}{3} (E_{+}^{1}E_{+}^{2} − \frac{1}{4})
= ±\frac{1}{3} (|00⟩⟨00| − \frac{1}{4}) ↔ ± \frac{1}{12} Diag(3, −1, −1, −1) .

(20)

The overall sign depends on whether we have a population excess or deficit in the ground state; for consistency, we shall generally assume the former. Observe that a unitary transformation of the density operator induces a transformation of the corresponding state vector just as it does for true pure states, since

Uρ̂_{00} Ũ = \frac{1}{3} ((U|00⟩) (U|00⟩)^{∼} − \frac{1}{4}) .  (21)

Similarly, because the NMR observables A = I_{x}^{k}, I_{y}^{k} are traceless, the ensembleaverage expectation value relative to a pseudo-pure density operator yields the ordinary expectation value versus the corresponding state vector:

Tr(Aρ̂_{00}) = \frac{1}{3} (Tr(A|00⟩⟨00|) − \frac{1}{4}Tr(A)) = \frac{1}{3}⟨00|A|00⟩  (22)

The general form of a pseudo-pure density operator is

ρ̂_{ψ} = \frac{N/2}{2^{N} −1} (|ψ⟩⟨ψ| − 2^{−N} ) ,  (23)

where |ψ⟩ is a normalized N-spin state vector, and the prefactor has been chosen so as to keep the maximum eigenvalue ∥ρ̂_{ψ}∥_{2} equal to that of the N-spin equilibrium density operator.

Even though we have defined them to have the same maximum eigenvalue, for N > 1 the remaining eigenvalues of ρ̂_{eq} and ρ̂_{ψ} differ, and hence there is no unitary transformation taking one to the other. There are nevertheless a number of nonunitary processes by which one can prepare pseudo-pure states. The most direct is to generate a spatially varying distribution of states across the sample, such that the ensemble average is pseudo-pure. This can be done by using a field gradient along the z-axis to create a position-dependent phase whose average is zero, thereby in effect setting the transverse (xy) components of the density operator to zero.^{2} For example, it is straightforward to show that the sequence

[\frac{π}{4}(I_{x}^{1} + I_{x}^{2})] → [πI_{z}^{1}I_{z}^{2}] → [−\frac{π}{6}(I_{y}^{1} + I_{y}^{2})]  (24)

applied to the two-spin equilibrium state ρ̂_{eq} yields

2^{− \frac{5}{2}} (\sqrt{3} (E_{+}^{1}E_{+}^{2} − \frac{1}{4} − I_{x}^{1}I_{x}^{2}) − I_{x}^{1}E_{−}^{2} − E_{−}^{1}I_{x}^{2}) ,  (25)

which is reduced by a z-gradient to (3/32)^{\frac{1}{2}} (E_{+}^{1}E_{+}^{2} − 1/4). Further RF and gradient pulse sequences which convert the equilibrium state of two and three

2In the homonuclear case, the zero-quantum coherences are not rapidly dephased by a z-gradient, so a slightly more complicated procedure is necessary.

spin systems to pseudo-pure states may be found in Ref. [14], but extending them to larger numbers of spins is not straightforward.

An alterative proposed by E. Knill et al. [25] is to “time-average” the results of several separate experiments. In the simple case of a two spin system, for example, the average of the three states

ρ̂_{123} ≡  \frac{1}{4}(I_{z}^{1} + I_{z}^{2}) ↔ \frac{1}{4} Diag(1, 0, 0, −1)
ρ̂_{231} ≡ \frac{1}{4}(I_{z}^{1} + 2I_{z}^{1}I_{z}^{2}) ↔ \frac{1}{4} Diag(1, 0, −1, 0)
ρ̂_{312} ≡ \frac{1}{4}(2I_{z}^{1}I_{z}^{2} + I_{z}^{2}) ↔ \frac{1}{4} Diag(1, −1, 0, 0)

(26)

is the pseudo-pure state

\frac{1}{3}(ρ̂_{123} + ρ̂_{231} + ρ̂_{312}) = \frac{1}{12}(2I_{z}^{1} + 2I_{z}^{2} + 4I_{z}^{1}I_{z}^{2})
↔ \frac{1}{12} Diag(3, −1, −1, −1) .

(27)

More generally, one can obtain the same results that one would get on a pseudo-pure state by averaging the results of the calculations over all 2^{N} − 1 cyclic permutations of the nonground state populations of the equilibrium density operator. Although this naive approach is not efficient, Knill et al. have shown that one can average over smaller groups in time O(N^{3}) with much the same effect.

A fundamentally different approach, first proposed by N. Gershenfeld & I. Chuang [18], involves working with subpopulations of the molecules distinguished by the states of additional ancilla spins. Gershenfeld & Chuang [10] have given an example of a two-spin conditional pseudo-pure state (as we call it), which is obtained by row/column permutation of the diagonal equilibrium density matrix Diag(3, 1, 1, −1, 1, −1, −1, −3)/16 of a three-spin system including one ancilla, namely

\frac{1}{16} Diag(3, −1, −1, −1, −3, 1, 1, 1)
↔ \frac{1}{16}(2I_{z}^{1}(2I_{z}^{2} + 2I_{z}^{3} + 4I_{z}^{2}I_{z}^{3}))
= \frac{1}{4}(E_{+}^{1} − E_{−}^{1})(E_{+}^{2}E_{+}^{3} − \frac{1}{4}) .

(28)

The last form makes it clear that in the subpopulation with the first spin “up”, which is labeled by E_{+}^{1}, and in the subpopulation with it “down”, which is labeled by E_{−}^{1}, spins 2 and 3 are in the pseudo-pure state E_{+}^{2}E_{+}^{3} − 1/4. Since the spectrum of spins 2 and 3 is antiphase with respect to the ancilla spin 1, one can select the subpopulations just by keeping only either positive or negative peaks. Although the situation is considerably more complicated with more spins, Gershenfeld and Chuang have shown that conditional pure states can be obtained (with some loss of signal) using as few as O(log(N)) ancillae; to our knowledge, the time-space-signal trade-offs of the various possibilities have not been fully worked out.

An alternative to conditional pure states, which we call relative pseudopure states, can be obtained via the partial trace operation (in NMR, decoupling [33]), rather than peak selection as above. For example, a two-spin relative pseudo-pure state is given by the partial trace over the ancilla spins 1 & 2 in

\begin{aligned}& \frac{1}{32} Diag(4, 2, 2, 0, 2, 0, −2, 0, 0, −2, 0, 2, 0, −2, −2, −4) \\ ↔ & \frac{1}{16} (E_{+}^{1}E_{+}^{2}(E_{+}^{3} + E_{+}^{4}) + E_{+}^{1}E_{−}^{2}(E_{+}^{3}E_{+}^{4} − E_{−}^{3}E_{+}^{4}) \\ & + E_{−}^{1}E_{+}^{2}(E_{−}^{3}E_{−}^{4} − E_{+}^{3}E_{−}^{4}) − E_{−}^{1}E_{−}^{2}(E_{−}^{3} + E_{−}^{4})) , \end{aligned}  (29)

which is again a permutation of the diagonal elements of ρ̂_{eq}. This can be seen by adding up the 4×4 blocks of the matrix, obtaining Diag(6, −2, −2, −2)/32. Alternatively, since the partial trace in the product operator formalism corresponds to simply eliminating those terms depending on either spins 1 or 2 and multiplying the remaining terms by 4 [34], we need only add up the multipliers of E_{+}^{1}E_{+}^{2}, …, E_{−}^{1}E_{−}^{2}, which yields

\begin{aligned}& \frac{1}{16}((1 + E_{+}^{3} − E_{−}^{3})(1 + E_{+}^{4} − E_{−}^{4}) − 1) \\ = & \frac{1}{4} (E_{+}^{3}E_{+}^{4} − \frac{1}{4}) ↔ \frac{1}{32} Diag(6, −2, −2, −2) . \end{aligned}  (30)

We now briefly consider the SNR (signal-to-noise ratio) of these methods of creating pseudo-pure states. It has been argued that since the equilibrium population of the ground state falls off exponentially with the number Figure 1: Negative base ten logarithm of the polarization P as a function of the logarithm of the ratio of the energy level spacing to k_{B}T for a one, four, sixteen and sixty-four spin pseudo-pure state obtained by cyclic averaging. For protons in a standard 500 MHz. spectrometer, P ∼ 10^{−5} at equilibrium. of spins, and all these methods are aimed in some fashion at isolating the signal from the ground state population, the SNR of all these methods must likewise decline exponentially with N [39]. Although this argument carries considerable weight, we shall see that the number and variety of the available methods renders the actual situation more complex. The standard to which the signal strength must be compared is that of a single spin in its equilibrium state, namely

ρ̂_{eq} = \frac{1}{2}I_{z}^{1} = \frac{1}{4}(|0⟩⟨0| − |1⟩⟨1|) .  (31)

The maximum eigenvalue ∥ρ̂_{eq}∥_{2} = 1/4 is what we will use as the standard signal strength for spins of like gyromagnetic ratio (as assumed throughout). We shall therefore calculate the SNR of a pseudo-pure state by transforming it to the corresponding ground state |0 ⋯ 0⟩⟨0 ⋯ 0|−2^{−N} (if need be), taking the partial trace over all but one of the spins, and multiplying the maximum eigenvalue of the result by 4.

The maximum eigenvalue of the partial trace over all but one of the spins in a pseudo-pure state obtained by cyclic averaging, as in Eq. (27), is easily seen to be N/(4(2^{N} − 1)), which decays almost exponentially with the number of spins N. There is an additional factor of \sqrt{2^{N} − 1} which comes from averaging over 2^{N} − 1 experiments, and gives a net SNR of N/(4\sqrt{2^{N} − 1}) for the average. The exponential time requirements of cyclic averaging will nonetheless force one to average over smaller groups, with consequently smaller improvements in the SNR. In any case, the SNR declines superpolynomially with N. Figure 1 shows how the signal strength changes as a function of the ratio of the energy level spacing to k_{B}T, relative to the signal in a perfectly polarized sample, when the pseudo-pure state is obtained by cyclic averaging, for varying numbers of spins.

Because of the many possible variations on the ideas and the difficulty of analyzing all of them, it is not practical to present simple formulae for the SNR of the other methods of preparing pseudo-pure states. Further complexity is added to the situation by the ability to combine the various methods above. A number of such combinations are given in Knill et al. [25], along with bounds on the SNR for each. In our laboratory we are developing a new method, again based on field gradients, which enables the sample to be divided into discrete volumes and separate unitary transformations to be applied to each. In principle, this permits multiple experiments to be performed, and their results added, in a single experiment, thereby performing an average over multiple experiments in constant time. This new method could also be used in a variety of combinations with existing methods.

It is nevertheless encouraging to observe that the SNR of the two-spin conditional and relative pseudo-pure states given in Eqs. (28) and (29) is 1/2 in both cases; this is exactly the decline in the ground state population of a two-spin system compared to a one-spin. In Eq. (28), we attain this “theoretical limit” because the expansion of the density operator consists of a single term conditioned on the state of a single ancilla; it is not possible to do as well with more spins. In Eq. (29), however, it is because such permutations are able to transfer polarization from one set of spins to another. We have found this makes it possible to derive a two-spin pseudo-pure state from a six-spin equilibrium state with no loss of SNR, whereas a simplistic groundstate population argument implies we should lose at least 1/2. This may be seen by adding up the rows in the rearrangement of ρ̂_{eq} shown in Eq. (32) below, which corresponds taking the traces of the four 16 × 16 blocks along the diagonal, and yields Diag(48, −16, −16, −16).

Diag \begin{matrix}( 6, & 4, & 4, & 4, & 4, & 4, & 4, & 2, & 2, & 2, & 2, & 2, & 2, & 2, & 2, & 2, \\ 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & −2, & −2, & −2, & −2, & −4, & −4, \\ 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & 0, & −2, & −2, & −2, & −2, & −4, & −4, \\ 2, & 2, & 2, & 2, & 2, & 2, & −2, & −2, & −2, & −2, & −2, & −2, & −2, & −4, & −4, & −6 & ) \end{matrix}

(32)

The partial trace over one of the two remaining spins then gives Diag(32, −32), which when divided by 128 (twice the partition function) yields \frac{1}{2}I_{z} as claimed.

A general algorithm has recently been given by Schulman & Vazirani [32] whereby one can “distill” an M-spin relative pure state from an ensemble of molecules each containing N spins. Starting from a net polarization per spin of ϵ, this algorithm yields M perfectly polarized spins providing M/N ∼ O(ϵ^{2}), a result anticipated by earlier work in NMR which showed that the amount of polarization that can be transferred to a single spin is at most O(\sqrt{N}) [35]. Unfortunately, given that ϵ ≈ 10^{−5} for protons at equilibrium in a standard 500 MHz. spectrometer, a molecule with of order 10^{10} spins would be needed to prepare a perfectly polarized state on a single spin — which is always in a pseudo-pure state! The importance of Schulman & Vazirani’s algorithm thus lies in the fact that it shows that it is at least theoretically possible to perform fully quantum computations with only polynomial overhead in space and time, starting from a thermal equilibrium state.

In summary, although no general and practical method of preparing pseudo-pure states with bounded SNR from high-temperature equilibrium states is currently known, it remains possible that such a method will be found. It is further worth pointing out that physical methods of “refrigerating” spins are available, for example optical pumping [29]. These are presently confined to very simple systems, but such a source of polarization could in principle be used in conjunction with polarization transfer techniques to produce (pseudo-)pure states on large numbers of spins. Whether or not NMR proves to be a suitable means of building a large-scale ensemble quantum computer in the future, some very significant tests of quantum mechanics and demonstrations of quantum computing are possible with the small numbers of quantum bits we can achieve today. To illustrate this, we will now present the results of NMR experiments which (with certain assumptions) disprove the existence of “hidden variables”.

4 MACROSCOPIC CONSEQUENCES OF QUAN-TUM CORRELATIONS

Given the success of the purely classical Bloch equations (and their multispin extensions) in describing the NMR phenomena [33], it is perhaps surprising that phenomena widely regarded as uniquely “quantum” are nonetheless manifest in macroscopic, room-temperature NMR spectroscopy. For example, Seth Lloyd has recently proposed that the nonclassical correlations in (Mermin’s version of) the GHZ state can be validated using NMR [27]. His approach involves using a fourth “observer” spin to perform a nondemolition measurement on the three spins in a GHZ state (or a pseudo-pure analogue thereof). Here we shall consider another, rather different way in which quantum correlations can be demonstrated by NMR. We stress at the outset that, because all the interacting spins involved reside in the same molecule, neither Lloyd’s approach nor ours can establish the nonlocality of the correlations.

The approach taken here was inspired by an educational paper published a few years ago, in which T. F. Jordan has shown that the contradictions with hidden variables implied by violations of Bell’s inequalities as well as by the GHZ and Hardy’s paradox can be derived entirely by consideration of the expectation values of product operators, rather than by observations on single spins [24]. This shows that, in principle, it is not necessary to use nondemolition measurements with an observer spin in order to perform experiments which demonstrate these contradictions by NMR; it can be done directly from observations on ensembles of the spins of interest, providing at least they are in a (pseudo-)pure state. In a companion paper to Jordan’s, N. D. Mermin points out that in real-life experiments it is nevertheless not possible to perform the measurements, either of single spins nor (by implication) of expectation values, with sufficient precision to establish the “perfect” (total) correlations on which “EPR” arguments against the existence of hidden variables are based [28]. In that same paper, however, Mermin shows that Hardy’s paradox is a consequence of the Clauser-Horne form of Bell’s inequality. This enables Hardy’s paradox [6, 21] to be extended to an open set in the Hilbert space of only two spins, to which the practically available experimental data can confine us.

In the following, we present the results of NMR experiments which implement the specific example of Hardy’s paradox presented by Mermin in an Appendix to his paper [28]. Let us map the “red” and “green” eigenstates |1G⟩ and |1R⟩ of Mermin’s measurement 1 to the spin states |0⟩ and |1⟩, respectively. It will be clearer here to relabel this measurement as “A”, and to use |α_{G}⟩ ≡ |0⟩ and |α_{R}⟩ ≡ |1⟩ as synonyms for its eigenbasis. Correspondingly, we will relabel Mermin’s measurement 2 as “B”, and denote its the eigenbasis by

|β_{G}⟩ ≡ \sqrt{\frac{3}{5}}|0⟩ − \sqrt{\frac{2}{5}}|1⟩ and |β_{R}⟩ ≡ \sqrt{\frac{2}{5}}|0⟩ + \sqrt{\frac{3}{5}}|1⟩ .  (33)

Then the state which Mermin has shown leads to a near-maximum violation of Bell’s inequality while also providing an example of Hardy’s paradox is

|ψ⟩ ≡ \frac{1}{2}|00⟩ + \sqrt{\frac{3}{8}}|01⟩ + \sqrt{\frac{3}{8}}|10⟩ .  (34)

To translate this into the context of NMR, we first note that the observable whose expectation value is the probability that measurement A yields the state |α_{G}⟩ is given by A ≡ E_{+} ≡ \frac{1}{2}(1+2I_{z}) (we drop the usual spin index because the measurements A & B are assumed the same for both spins). Similarly the observable which gives the probability that measurement B yields |β_{G}⟩ is

\begin{aligned}B ≡ |β_{G}⟩⟨β_{G}| & = & \frac{3}{5}|0⟩⟨0| + \frac{2}{5}|1⟩⟨1| − \sqrt{\frac{6}{25}}(|0⟩⟨1| + |1⟩⟨0|) \\ & = & \frac{1}{2} + \frac{1}{5}I_{z} − \sqrt{\frac{24}{25}}I_{x} . \end{aligned}  (35)

In addition, the density operator (including the identity) of Mermin’s state is

Ψ ≡ |ψ⟩⟨ψ| = \frac{1}{4} + \frac{1}{8} (I_{z}^{1} + I_{z}^{2}) − \frac{1}{2}I_{z}^{1}I_{z}^{2}
+ \sqrt{\frac{3}{8}} (I_{x}^{1}(1 + 2I_{z}^{2}) + (1 + 2I_{z}^{1})I_{x}^{2})
+ \frac{3}{4} (I_{x}^{1}I_{x}^{2} + I_{y}^{1}I_{y}^{2}) .

(36)

The state |01⟩ is obviously related to |00⟩ by a rotation of spin 2, while |00⟩ can likewise by rotated to |10⟩, but without affecting |01⟩, by a conditional rotation of spin 1. We shall denote these by

P (ϕ) ≡ e^{−ıϕI_{y}^{2}} and Q(θ) ≡ e^{−ıθI_{y}^{1}E_{+}^{2}} ,  (37)

respectively. They act consecutively on the ground state to yield

⟨00|P̃ (ϕ)Q̃(θ) = [ cos(θ/2) cos(ϕ/2), sin(θ/2) cos(ϕ/2), sin(ϕ/2), 0 ] ,

(38)

which is easily verified to equal ⟨ψ| = [1/2, \sqrt{3/8}, \sqrt{3/8}, 0] when

ϕ = 2 arctan(\sqrt{3/5}) and θ = 2 arctan(\sqrt{3/2}) .  (39)

Using the product operator techniques presented in section 2, these transformations are readily implemented by NMR pulse sequences.

The next thing to notice is that if we take expectation values with the usual idempotents E_{+}^{1}E_{+}^{2}, …, E_{−}^{1}E_{−}^{2}, we get

\frac{1}{4} = 4 ⟨ΨE_{+}^{1}E_{+}^{2}⟩ ≡ 4 ⟨ΨA^{1}A^{2}⟩
\frac{3}{8} = 4 ⟨ΨE_{+}^{1}E_{−}^{2}⟩ ≡ 4 ⟨ΨA^{1}(1 − A^{2})⟩
\frac{3}{8} = 4 ⟨ΨE_{−}^{1}E_{+}^{2}⟩ ≡ 4 ⟨Ψ(1 − A^{1})A^{2}⟩
0 = 4 ⟨ΨE_{−}^{1}E_{−}^{2}⟩ ≡ 4 ⟨Ψ(1 − A^{1})(1 − A^{2})⟩ .

(40)

These correspond to the diagonal of the density matrix in the usual I_{z} basis,

diag(Ψ) = [\frac{1}{4}, \frac{3}{8}, \frac{3}{8}, 0] (A on 1, A on 2) ,  (41)

which contains the probabilities of the four possible outcomes of performing measurement A on both spins (as shown).

The product operator form of B immediately makes clear that measurement B is nothing more or less than the measurement of the magnetization of the spin along an axis inclined at an angle of ζ ≡ arctan(\sqrt{24}) = π − θ to the z-axis in the xz-plane. Letting R ≡ exp(−ıζI_{y}), it follows that the probability that measurement B on spin 1 yields “G” (i.e. |β_{G}⟩) is

4⟨ΨB^{1}⟩ = 4⟨ΨR̃^{1}A^{1}R^{1}⟩ = 4⟨R^{1}ΨR̃^{1}A^{1}⟩ .  (42)

with a similar expression for spin 2. More generally, the probabilities of the outcomes of the other combinations of measurements are given by the diagonals of the transformed density matrices:

diag (R^{2}ΨR̃^{2}) = [0, \frac{5}{8}, \frac{9}{40}, \frac{3}{20}]  (A on 1, B on 2)
diag (R^{1}ΨR̃^{1}) = [0, \frac{9}{40}, \frac{5}{8}, \frac{3}{20}]  (B on 1, A on 2)
diag (R^{1}R^{2}ΨR̃^{2}R̃^{1}) = [\frac{9}{100}, \frac{27}{200}, \frac{27}{200}, \frac{16}{25}] (B on 1, B on 2)

(43)

For compactness, let us denote these probabilities by Ψ_{kl}^{ij}, where i, j ∈ {A, B} are the measurements and k, l ∈ {G, R} are the corresponding outcomes, e.g. Ψ_{GR}^{AB} = 4⟨ΨA^{1}(1 − B^{2})⟩.

We may translate Mermin’s proof [28] that these probabilities are incompatible with hidden variables associated with the individual spins into this context as follows: First, since Ψ_{GG}^{AB} = Ψ_{GG}^{BA} = 0, in any molecule wherein one of the spins is parallel to the z-axis the other must be antiparallel to the axis of measurement B and vice versa. Hence, since Ψ_{GG}^{BB} is nonzero, in some molecules (9%, to be precise) both spins must be antiparallel to the z-axis. But this contradicts the fact that Ψ_{RR}^{AA} = 0. More generally, Mermin has shown that

Ψ_{GG}^{BB} ≤ Ψ_{GG}^{AB} + Ψ_{RR}^{AA} + Ψ_{GG}^{BA}  (44)

is an example of the Clauser-Horne form of Bell’s inequality. Thus to disprove the existence of such one-particle hidden variables it is sufficient to determine these probabilities to ±2% or so.

At this point we encounter a significant complication, which is that the “strong” (von Neumann) measurements assumed in their analyses by Jordan and Mermin cannot be implemented by NMR; we can only perform “weak” (nonperturbing) measurements of the relative population differences between states connected by single spin flips [14]. This is done by applying a magnetic field gradient along the z-axis, which (as previously described) dephases any transverse components in the density operator. Thereafter, a pair of “soft” π/2 readout pulses, each tuned to the frequency of just one of the two spins, produces a pair of spectra each with two peaks whose heights are proportional to the population differences between pairs of states connected by flips of that spin. The factor relating the peak heights to the corresponding differences in the probabilities of the states can be determined from spectra collected on the pseudo-pure ground state, after which it is straightforward to convert the differences into the corresponding absolute probabilities by linear least squares, subject to the constraint that their sum is unity. We shall encounter field gradients again in the next section, when we show how they can also be used to implement precisely controlled decoherence models.

Thus the overall experiment consists of collecting ten spectra, as follows:

1. Prepare the state Ψ, by first preparing the pseudo-pure ground state |00⟩ using one of the previously described methods, and then transforming it by Q(θ)P (ϕ).

2. Use a selective radio-frequency pulse to apply the rotation R to those spins on which measurement B is to be performed.

3. Use a z-gradient to dephase the transverse components of the resulting density operator.

4. Apply a readout pulse to one of the spins, and collect the corresponding spectrum; repeat steps 1 – 3 and then do the same for the other spin.

5. Repeat steps 1 - 4 for each of the four combinations of measurements AA, AB, BA and BB on the two spins.

6. Collect two additional calibration spectra by applying soft readout pulses to each spin in the pseudo-pure ground state.

A^{C}A^{H}  A^{C}B^{H} B^{C}A^{H}  B^{C}B^{H}

Figure 2: The pairs of ^{13}C-labeled chloroform spectra (carbon left, proton right) obtained by performing the four combinations of measurements A^{C}A^{H}, A^{C}B^{H}, B^{C}A^{H} and B^{C}B^{H} on the pseudo-pure form of Mermin’s state Ψ. The spectra have been normalized by the height of the peak of the corresponding spin in the pseudo-pure ground state, and the horizontal axis is in kHz. The transitions of the peaks, from left to right, are |0^{C}0^{H}⟩ ↔ |1^{C}0^{H}⟩, |0^{C}1^{H}⟩ ↔ |1^{C}1^{H}⟩, |0^{C}0^{H}⟩ ↔ |0^{C}1^{H}⟩, |1^{C}0^{H}⟩ ↔ |1^{C}1^{H}⟩.

Probabilities of Outcomes G & R for the Measurements A & B (Carbon, Proton) Demonstrating Hardy’s Paradox, as Derived from the Chloroform NMR Spectra in Fig. 2

Measurements  (G, G)  (G, R)  (R, G)  (R, R)  Residuals
(A^{C}, A^{H})  0.253  0.380  0.366  0.001  0.008
(A^{C}, B^{H})  0.029  0.609  0.217  0.145  0.018
(B^{C}, A^{H})  −0.002  0.230  0.614  0.159  0.005
(B^{C}, B^{H})  0.097  0.125  0.156  0.622  0.021

These experiments were performed on a Bruker 400 MHz. spectrometer using the two spin \frac{1}{2} nuclei in ^{13}C-labeled chloroform.

The spectra obtained from steps 1 – 5 of this experiment are shown in Fig. 2. The probabilities derived from these peak heights, and the residual associated with each, are shown in the table below. It follows that Bell’s inequality is violated by

Ψ_{GG}^{AB} + Ψ_{RR}^{AA} + Ψ_{GG}^{BA} − Ψ_{GG}^{BB}
= 0.029 + 0.001 − 0.002 − 0.097 = − 0.069 .

(45)

A rigorous error analysis is not possible, because the dominant errors in NMR spectra (e.g. RF field inhomogeneity) are not statistical. If we nevertheless take the mean RMS residual (half the total residuals shown in the table) of 0.0065 as an estimate of the errors and assume they are independent between spectra, the expected error in the sum of these four numbers is only 0.013, so that this violation of Bell’s inequality appears significant.

It is important to point out that, because these measurements are performed on a weakly polarized pseudo-pure state rather than on a true pure state, they are subject to interpretation. If one were to pull the molecules out of the sample one at a time and perform the measurements A and B with a Stern-Gerlach apparatus, the resulting probabilities would all be very close to 1/4, which would not violate Bell’s inequality. This is because the vast majority of the density operator is contained in its identity component, which however does not contribute to the NMR spectra.

Indeed, a recent preprint by Braustein, Caves, Jozsa, Linden, Popescu and Shack claims that since the total density operator of such a weakly polarized sample can be explained by ensemble of unentangled spin systems, such experiments should be regarded as “simulations” of true quantum systems and not instances thereof [7]. Nevertheless, this is not the only valid definition of mixed state entanglement (as Braustein et al. also point out). For example, it is well-known that one can teleport an unknown mixed state with better fidelity than could be achieved by local operations and classical communication, even though the state does not violate any Bell inequality (see exercise 5.5 in [31]).

In contrast to Braustein et al.’s point of view, our claim to have obtained a violation of Bell’s inequality is based on the assumption that the spins of each molecule in a pseudo-pure ensemble are in a definite, albeit unknown, pure state, and that the molecules contributing to the spectra of such an ensemble are all in the same corresponding pure state. The arbitrary choice of ensemble for a density operator is in many respects similar to the arbitrary choice of a coordinate system in physical problems more generally, and our ensemble interpretation for that part of a pseudo-pure density operator which can be observed by NMR is arguably the simplest, in that it involves the smallest possible number of pure states: one. It should further be clear that, because the unobserved molecules are in a different state from those which contribute to the spectra, this interpretation of our experiments is quite different from the “loophole” in the Aspect experiments [1], where the unobserved photon pairs are expected to follow the same statistics as the observed ones.

This microscopic interpretation of pseudo-pure density operators, which in one way or another underlies all existing claims that it is possible to observe quantum phenomena by macroscopic NMR spectroscopy, will be further justified in future publications. For now, we will simply point out that the interpretation is at least supported by the otherwise incredible ability of NMR spectroscopy on pseudo-pure states to reproduce all manner of quantum phenomena. To further emphasize this, we will now present experimental results which demonstrate the principles of quantum error correction from an NMR perspective.

5 QUANTUM ERROR CORRECTION BY NMR SPECTROSCOPY

The error correcting code we have chosen to illustrate by NMR is well-known in the field [26], and uses two ancilla (labeled 2 & 3) to encode the state of a data spin (labeled 1). Letting S^{2|1} and S^{3|1} be c-NOT’s, and R_{90}^{123} ≡ exp(−ı\frac{π}{2}(I_{y}^{1} + I_{y}^{2} + I_{y}^{3})), the encoding operation proceeds as follows:

\begin{matrix}(α|0⟩ + β|1⟩)|00⟩ ⟶^{S^{2|1}}⟶^{S^{3|1}}⟶^{R_{90}^{123}} α|+++⟩ + β|−−−⟩ \\ (where |±±±⟩ ≡ (|0⟩ ± |1⟩)(|0⟩ ± |1⟩)(|0⟩ ± |1⟩)) \end{matrix}  (46)

Decoding consists of applying the inverse operations in the reverse order, which acts on the states obtained by single sign-flip errors as follows:

\begin{matrix}α|++−⟩ + β|−−+⟩ ⟶^{R_{−90}^{123}}⟶^{S^{3|1}}⟶^{S^{2|1}} (α|0⟩ + β|1⟩)|01⟩ \\ α|+−+⟩ + β|−+−⟩ ⟶^{R_{−90}^{123}}⟶^{S^{3|1}}⟶^{S^{2|1}} (α|0⟩ + β|1⟩)|10⟩ \\ α|−++⟩ + β|+−−⟩ ⟶^{R_{−90}^{123}}⟶^{S^{3|1}}⟶^{S^{2|1}} (α|1⟩ + β|0⟩)|11⟩ \end{matrix}  (47)

It follows that a Toffli gate T^{1|23}, which flips the data spin conditional on the ancillae being in the state |11⟩, will correct a sign-flip error in the data spin and leave it alone otherwise, even if an error occurs in the ancillae.

In practice, errors in quantum computers are not expected to be single sign-flips, but rather small random phase errors which cumulatively result in decoherence. Nevertheless, we can show that the ability to correct sign-flips implies the ability to cancel the effect of such phase errors to first order.

Random phase errors correspond to the propagator exp(−ı(χ^{1}I_{z}^{1} + χ^{2}I_{z}^{2} + χ^{3}I_{z}^{3})), which acts to first order on the encoded state as:

\begin{aligned}& exp(−ı(χ^{1}I_{z}^{1} + χ^{2}I_{z}^{2} + χ^{3}I_{z}^{3}) (α|+++⟩ + β|−−−⟩) \\ ≈ & (α|+++⟩ + β|−−−⟩) − ıχ^{1} (α|−++⟩ + β|+−−⟩) \\ & − ıχ^{2} (α|+−+⟩ + β|−+−⟩) − ıχ^{3} (α|++−⟩ + β|−−+⟩) \end{aligned} (48)

Since decoding and the error-correcting Toffoli gate are likewise linear, it follows that the first-order effects of phase errors are cancelled as claimed. Note this argument makes no assumptions concerning the correlations among the errors!

Experimental results demonstrating these expectations have recently been published [13]. In the following, we shall present a more detailed explanation of how the error correction works using the product operator formalism, along with selected experimental data illustrating and validating this explanation. We shall assume that the data spin is in one of the states 1 (unpolarized), I_{x}^{1}, I_{y}^{1} or I_{z}^{1}. Although these are mixed states, each consists of an incoherent sum of pure states, e.g. 2I_{z}^{1} = |0⟩⟨0| − |1⟩⟨1|, so if error correction works on these pure states, by linearity it will also work on the mixtures (and vice versa). In these terms, a complete set of initial states ρ̂_{A} for error correction are:

\begin{matrix}E & {2 \atop +} & E & {3 \atop +} \\ I & {1 \atop x} & E & {2 \atop +} & E & {3 \atop +} \\ I & {1 \atop y} & E & {2 \atop +} & E & {3 \atop +} \\ I & {1 \atop z} & E & {2 \atop +} & E & {3 \atop +} \end{matrix} \} ≡ ρ̂_{A}^{1}E_{+}^{2}E_{+}^{3} = ρ̂_{A}  (49)

The corresponding states ρ̂_{B} to which they are mapped by encoding are:

ρ̂_{B} ≡ \{ \begin{matrix}\frac{1}{4} + I_{x}^{1}I_{x}^{2} + I_{x}^{1}I_{x}^{3} + I_{x}^{2}I_{x}^{3} \\ I_{z}^{1}I_{y}^{2}I_{y}^{3} + I_{y}^{1}I_{z}^{2}I_{y}^{3} + I_{y}^{1}I_{y}^{2}I_{z}^{3} − I_{z}^{1}I_{z}^{2}I_{z}^{3} \\ I_{z}^{1}I_{z}^{2}I_{y}^{3} + I_{z}^{1}I_{y}^{2}I_{z}^{3} + I_{y}^{1}I_{z}^{2}I_{z}^{3} − I_{y}^{1}I_{y}^{2}I_{y}^{3} \\ \frac{1}{4}(I_{x}^{1} + I_{x}^{2} + I_{x}^{3}) + I_{x}^{1}I_{x}^{2}I_{x}^{3} \end{matrix}  (50)

We note the last three states in Eq. (49) can be prepared (with a 50% loss of polarization) from the average of twice Eq. (28) with

\frac{1}{16} Diag(3, 1, 1, 1, −3, −1, −1, −1)
↔ \frac{1}{16}(I_{z}^{1}(3 + 2I_{z}^{2} + 2I_{z}^{3} + 4I_{z}^{2}I_{z}^{3}))
= \frac{1}{4}(E_{+}^{1} − E_{−}^{1})(E_{+}^{2}E_{+}^{3} + \frac{1}{2}) .

(51)

In NMR, decoherence occurs principally through the randomly fluctuating external magnetic fields at each spin k along the z-axis, B_{z}^{k}. The effect of these fields is most simply described in the spherical operator basis 1, I_{z}^{k} and I_{±}^{k} ≡ I_{x}^{k} ± ıI_{y}^{k}, as opposed to the Cartesian basis used up to now. The products of these basis elements can be shown to decay exponentially at rates proportional to the mean-square field \overline{(B_{z}^{k})^{2}} for I_{±}^{k} (as well as I_{±}^{k}I_{z}^{ℓ}, I_{±}^{k}I_{z}^{ℓ}I_{z}^{m}), and to

\overline{(B_{z}^{k} − B_{z}^{ℓ})^{2}}  for I_{+}^{k}I_{−}^{ℓ} & I_{−}^{k}I_{+}^{ℓ},
\overline{(B_{z}^{k} + B_{z}^{ℓ})^{2}}  for I_{+}^{k}I_{+}^{ℓ} & I_{−}^{k}I_{−}^{ℓ},
\overline{(B_{z}^{k} + B_{z}^{ℓ} − B_{z}^{m})^{2}} for I_{+}^{k}I_{+}^{ℓ}I_{−}^{m} & I_{−}^{k}I_{−}^{ℓ}I_{+}^{m}, etc.,
and \overline{(B_{z}^{k} + B_{z}^{ℓ} + B_{z}^{m})^{2}} for I_{+}^{k}I_{+}^{ℓ}I_{+}^{m} & I_{−}^{k}I_{−}^{ℓ}I_{−}^{m}.

(52)

These products are referred to as single (SQC1: I_{±}^{k}), zero (ZQC: I_{±}^{k}I_{∓}^{ℓ}), double (DQC: I_{±}^{k}I_{±}^{ℓ}), three-spin single (SQC3: I_{±}^{k}I_{±}^{ℓ}I_{∓}^{m}, etc.) and triple (TQC: I_{±}^{k}I_{±}^{ℓ}I_{±}^{m}) quantum coherences, respectively.

We shall consider two extreme forms of decoherence. In the first, the fields at the different spins are uncorrelated, and hence the random variables χ^{k} can be assumed to be identically distributed and independent. In the second, they are assumed to be totally correlated. By Eq. (52), the relative rates of decoherence in these two cases are:

ZQC SQC1 SQC3 DQC TQC
Uncorrelated:  2  1  3  2  3
Totally Correlated: 0  1  1  4  9

(53)

Decomposing ρ̂_{B} into a spherical basis, multiplying by decaying exponentials with the above rates normalized by the SQC1 decay rate τ, and returning to the Cartesian basis gives

ρ̂_{C} ≡ \{ \begin{matrix}\frac{1}{4} + (I_{x}^{1}I_{x}^{2} + I_{x}^{1}I_{x}^{3} + I_{x}^{2}I_{x}^{3})e^{−2t/τ} \\ (I_{z}^{1}I_{y}^{2}I_{y}^{3} + I_{y}^{1}I_{z}^{2}I_{y}^{3} + I_{y}^{1}I_{y}^{2}I_{z}^{3})e^{−2t/τ} − I_{z}^{1}I_{z}^{2}I_{z}^{3} \\ (I_{z}^{1}I_{z}^{2}I_{y}^{3} + I_{z}^{1}I_{y}^{2}I_{z}^{3} + I_{y}^{1}I_{z}^{2}I_{z}^{3})e^{−t/τ} − I_{y}^{1}I_{y}^{2}I_{y}^{3}e^{−3t/τ} \\ \frac{1}{4}(I_{x}^{1} + I_{x}^{2} + I_{x}^{3})e^{−t/τ} + I_{x}^{1}I_{x}^{2}I_{x}^{3}e^{−3t/τ} \end{matrix}  (54)

in the uncorrelated case, and

ρ̂_{C} ≡ \{ \begin{matrix}\frac{1}{4} + \frac{1}{2}(I_{x}^{1}I_{x}^{2} + I_{x}^{1}I_{x}^{3} + I_{x}^{2}I_{x}^{3} + I_{y}^{1}I_{y}^{2} + I_{y}^{1}I_{y}^{3} + I_{y}^{2}I_{y}^{3}) \\ + \frac{1}{2}(I_{x}^{1}I_{x}^{2} + I_{x}^{1}I_{x}^{3} + I_{x}^{2}I_{x}^{3} − I_{y}^{1}I_{y}^{2} − I_{y}^{1}I_{y}^{3} − I_{y}^{2}I_{y}^{3})e^{−4t/τ} \\ \frac{1}{2}(I_{z}^{1}(I_{x}^{2}I_{x}^{3} + I_{y}^{2}I_{y}^{3}) + I_{z}^{2}(I_{x}^{1}I_{x}^{3} + I_{y}^{1}I_{y}^{3}) + I_{z}^{3}(I_{x}^{1}I_{x}^{2} + I_{y}^{1}I_{y}^{2})) \\ − \frac{1}{2}(I_{z}^{1}(I_{x}^{2}I_{x}^{3} − I_{y}^{2}I_{y}^{3}) + I_{z}^{2}(I_{x}^{1}I_{x}^{3} − I_{y}^{1}I_{y}^{3}) + I_{z}^{3}(I_{x}^{1}I_{x}^{2} − I_{y}^{1}I_{y}^{2})) \\ e^{−4t/τ} − I_{z}^{1}I_{z}^{2}I_{z}^{3} \\ (I_{z}^{1}I_{z}^{2}I_{y}^{3} + I_{z}^{1}I_{y}^{2}I_{z}^{3} + I_{y}^{1}I_{z}^{2}I_{z}^{3})e^{−t/τ} \\ + \frac{1}{4}(I_{x}^{1}I_{x}^{2}I_{y}^{3} + I_{x}^{1}I_{y}^{2}I_{x}^{3} + I_{y}^{1}I_{x}^{2}I_{x}^{3} + 3I_{y}^{1}I_{y}^{2}I_{y}^{3})e^{−t/τ} \\ − \frac{1}{4}(I_{x}^{1}I_{x}^{2}I_{y}^{3} + I_{x}^{1}I_{y}^{2}I_{x}^{3} + I_{y}^{1}I_{x}^{2}I_{x}^{3} − I_{y}^{1}I_{y}^{2}I_{y}^{3})e^{−9t/τ} \\ \frac{1}{4}(I_{x}^{1} + I_{x}^{2} + I_{x}^{3})e^{−t/τ} \\ + \frac{1}{4}(3I_{x}^{1}I_{x}^{2}I_{x}^{3} + I_{y}^{1}I_{y}^{2}I_{x}^{3} + I_{y}^{1}I_{x}^{2}I_{y}^{3} + I_{x}^{1}I_{y}^{2}I_{y}^{3})e^{−t/τ} \\ + \frac{1}{4}(I_{x}^{1}I_{x}^{2}I_{x}^{3} − I_{y}^{1}I_{y}^{2}I_{x}^{3} − I_{y}^{1}I_{x}^{2}I_{y}^{3} − I_{x}^{1}I_{y}^{2}I_{y}^{3})e^{−9t/τ} \end{matrix}

(55) in the totally correlated. The decoding operation converts this to

ρ̂_{D} ≡ \{ \begin{matrix}\frac{1}{4} + (I_{z}^{2} + I_{z}^{3} + I_{z}^{2}I_{z}^{3})e^{−2t/τ} \\ (\frac{1}{2}I_{x}^{1}I_{z}^{2} + \frac{1}{2}I_{x}^{1}I_{z}^{3} + I_{x}^{1}I_{z}^{2}I_{z}^{3})e^{−2t/τ} + \frac{1}{4}I_{x}^{1} \\ (\frac{1}{4}I_{y}^{1} + \frac{1}{2}I_{y}^{1}I_{z}^{2} + \frac{1}{2}I_{y}^{1}I_{z}^{3})e^{−t/τ} + I_{y}^{1}I_{z}^{2}I_{z}^{3}e^{−3t/τ} \\ (\frac{1}{4}I_{z}^{1} + \frac{1}{2}I_{z}^{1}I_{z}^{2} + \frac{1}{2}I_{z}^{1}I_{z}^{3})e^{−t/τ} + I_{z}^{1}I_{z}^{2}I_{z}^{3}e^{−3t/τ} \end{matrix}  (56)

in the uncorrelated case, and

ρ̂_{D} ≡ \{ \begin{matrix}\frac{1}{4} + \frac{1}{2}(\frac{1}{2}I_{z}^{2} + \frac{1}{2}I_{z}^{3} + I_{z}^{2}I_{z}^{3} − 2I_{x}^{1}I_{z}^{2}I_{y}^{3} − 2I_{x}^{1}I_{y}^{2}I_{z}^{3} + I_{y}^{2}I_{y}^{3}) \\ + \frac{1}{2}(\frac{1}{2}I_{z}^{2} + \frac{1}{2}I_{z}^{3} + I_{z}^{2}I_{z}^{3} + 2I_{x}^{1}I_{z}^{2}I_{y}^{3} + 2I_{x}^{1}I_{y}^{2}I_{z}^{3} − I_{y}^{2}I_{y}^{3})e^{−4t/τ} \\ \frac{1}{2}(I_{x}^{1}(I_{y}^{2}I_{y}^{3} + I_{z}^{2}I_{z}^{3}) + \frac{1}{2}I_{z}^{3}(I_{x}^{1} − I_{x}^{2}) + \frac{1}{2}I_{z}^{2}(I_{x}^{1} − I_{x}^{3})) + \frac{1}{4}I_{x}^{1} \\ + \frac{1}{2}(I_{x}^{1}(I_{y}^{2}I_{y}^{3} − I_{z}^{2}I_{z}^{3}) + \frac{1}{2}I_{z}^{3}(I_{x}^{1} + I_{x}^{2}) + \frac{1}{2}I_{z}^{2}(I_{x}^{1} + I_{x}^{3}))e^{−4t/τ} \\ (\frac{1}{2}I_{y}^{1}I_{z}^{3} + \frac{1}{2}I_{y}^{1}I_{z}^{2} + \frac{1}{4}I_{y}^{1})e^{−t/τ} \\ + \frac{1}{4}(I_{z}^{1}I_{z}^{2}I_{y}^{3} + I_{z}^{1}I_{y}^{2}I_{z}^{3} − I_{y}^{1}I_{y}^{2}I_{y}^{3} − 3I_{y}^{1}I_{z}^{2}I_{z}^{3})e^{−t/τ} \\ − \frac{1}{4}(I_{z}^{1}I_{z}^{2}I_{y}^{3} + I_{z}^{1}I_{y}^{2}I_{z}^{3} − I_{y}^{1}I_{y}^{2}I_{y}^{3} + I_{y}^{1}I_{z}^{2}I_{z}^{3})e^{−9t/τ} \\ \frac{1}{4}(I_{z}^{1} + 2I_{z}^{1}I_{z}^{2} + 2I_{z}^{1}I_{z}^{3})e^{−t/τ} \\ + \frac{1}{4}(3I_{z}^{1}I_{z}^{2}I_{z}^{3} + I_{y}^{1}I_{y}^{2}I_{z}^{3} + I_{y}^{1}I_{z}^{2}I_{y}^{3} + I_{z}^{1}I_{y}^{2}I_{y}^{3})e^{−t/τ} \\ + \frac{1}{4}(I_{z}^{1}I_{z}^{2}I_{z}^{3} − I_{y}^{1}I_{y}^{2}I_{z}^{3} − I_{y}^{1}I_{z}^{2}I_{y}^{3} − I_{z}^{1}I_{y}^{2}I_{y}^{3})e^{−9t/τ} \end{matrix} (57)

in the totally correlated. This is clearly getting a little messy, and it gets much worse after the Toffoli gate! Therefore, we shall only present the partial trace over the ancillae after applying the Toffoli, which is

ρ̂_{E}^{1} ≡ \{ \begin{matrix}1 \\ I & {1 \atop x} \\ I & {1 \atop y} & (\frac{3}{2}e^{−t/τ} − \frac{1}{2}e^{−3t/τ} ) \\ I & {1 \atop z} & (\frac{3}{2}e^{−t/τ} − \frac{1}{2}e^{−3t/τ} ) \end{matrix}  (58)

in the uncorrelated case, and

ρ_{E}^{1} ≡ \{ \begin{matrix}1 \\ I & {1 \atop x} \\ I & {1 \atop y} & (\frac{3}{2}e^{−t/τ} − \frac{3}{8}e^{−t/τ} − \frac{1}{8}e^{−9t/τ} ) \\ I & {1 \atop z} & (\frac{3}{2}e^{−t/τ} − \frac{3}{8}e^{−t/τ} − \frac{1}{8}e^{−9t/τ} ) \end{matrix}  (59)

in the correlated. The slope of these curves at t = 0 is zero in all cases, as expected.

In order to demonstrate these results by NMR solution-state spectroscopy, a precise implementation of the above decoherence models is needed. This was achieved by using gradient-diffusion methods. In these methods, a magnetic field gradient is created along the z-axis; as previously described, this dephases the transverse (xy) magnetization. More precisely, a field gradient causes the transverse magnetization to precess at rates which depend linearly on its z-coordinate, thereby winding it into a spiral about the z-axis whose average transverse magnetization is essentially zero. The gradient is turned off for a given time interval t, during which diffusion of the molecules along z blurs the spiral. The gradient is then reversed, causing the magnetization to refocus and so create an “echo”. Because those molecules which have moved now precess at a different rate, their magnetization is not refocussed, so the magnitude of the echo decays exponentially with t. Because all the spins in each molecule are subject to the same change in field, this constitutes a true implementation of the totally correlated model. Using radio-freqency gradients, it is also possible to dephase each spin separately, thereby implementing the uncorrelated model. At this time, however, we have collected and processed data only for the ρ̂_{A}^{1} = I_{z}^{1} state with the totally correlated model.

Although it is possible to prepare the state I_{z}^{1}E_{+}^{2}E_{+}^{3} as noted above, we have chosen to illustrate the above analysis by preparing the states I_{z}^{1}, 2I_{z}^{1}I_{z}^{2}, 2I_{z}^{1}I_{z}^{3} and 4I_{z}^{1}I_{z}^{2}I_{z}^{3} in four separate experiments, each using sixteen different decoherence times t. Because the SQC and TQC contributing to 4I_{z}^{1}I_{z}^{2}I_{z}^{3} Figure 3: Experimental NMR data illustrating the decay of each of the product operators I_{z}^{1}, 2I_{z}^{1}I_{z}^{2}, 2I_{z}^{1}I_{z}^{3}, 4I_{z}^{1}I_{z}^{2}I_{z}^{3}, as functions of the time allowed for gradient diffusion (see text), together with the least-squares fits to their logarithms. The single and triple quantum coherences in 4I_{z}^{1}I_{z}^{2}I_{z}^{3}, (negative curves) have been plotted and fit separately. The sum of these data and the fits are also shown (topmost curve), which illustrates that error correction cancels the decay of the encoded state to first order as expected. refocussed at different times, this further enabled us to follow their evolutions separately. The results of these experiments are plotted against the time t in Figure 2, along with the corresponding logarithmic fits. It may be seen that the sum of the data and the fits thereto (also shown) do indeed exhibit a near-zero initial slope, in accord with the above calculations. Our published report [13] includes the results of further experiments (performed by E. Knill and R. Laflamme) with the natural and far more complicated decoherence processes that occur in solution. These are more difficult to interpret, but are nevertheless consistent with the state preservation expected from error correction. Additional experiments and more detailed calculations are in progress.

While a method of inhibiting decoherence (T_{2} relaxation) during NMR pulse sequences would be highly desirable, there are strong reasons to doubt that quantum error correction will be useful in this regard. First, the ancillae must be placed in a pseudo-pure state, which as we have shown above entails a loss of 50% of the signal for each “data” spin; this is more than is recovered by error correction. In addition, the ancillae must be returned to a pseudopure state uncorrelated with the state of the data spin(s), or else “fresh” ancillae in such a state must be continuously available, in order to inhibit decoherence over an appreciable period of time by the repeated correction of errors. Nevertheless, we feel that the basic idea underlying error correction of preparing multiple quantum coherences, allowing them to decoher, and then mixing them so as to determine their relative rates of relaxation, may be of considerable use in NMR studies of the statistics of molecular motion. This in turn is one of the most important applications of NMR spectroscopy. Conversely, whereas NMR spectroscopists have previously used their methods solely to unravel the secrets of naturally occurring systems, it now appears possible to use these same methods to engineer artificial systems in which the basic principles of quantum mechanics and the emergence of the classical world through decoherence can be studied in unprecedented detail [20].

ACKNOWLEDGEMENTS

We thank E. Knill and R. Laflamme of Los Alamos National Labs for teaching us about quantum error correcting codes, and S. Braunstein and R. Jozsa for useful discussions on mixed-state correlation. This work was supported by the U. S. Army Research Office under grant number DAAG 55-97-1-0342 from the DARPA Ultrascale Computing Program.

References

[1] A. Aspect, J. Dalibard, G. Roger. Experimental tests of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett., 49:1804–1807, 1982.

[2] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. Elementary gates for quantum computation. Phys. Rev. A, 52:3457–3467, 1995.

[3] K. Blum. Density Matrix Theory and Applications (2nd ed.). Plenum Press, New York City, 1996.

[4] B. Boulat and M. Rance. Algebraic formulation of the product operator formalism in the numerical simulation of the dynamic behavior of multispin systems. Mol. Phys., 83:1021–1039, 1994.

[5] M. Boyer, G. Brassard, P. Hoyer, and A. Tapp. Tight bounds on quantum searching. Fort. Phys., 46:493–505, 1998.

[6] D. Branning. Does nature violate local realism? Am. Sci., 85:160–167, 1997.

[7] S. L. Braustein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu and R. Schack. Separability of very noisy mixed states and implications for NMR quantum computing. LANL preprint quant-ph/9811018.

[8] I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd. Experimental realization of a quantum algorithm. Nature, 393:143–146, 1998.

[9] I. L. Chuang, N. A. Gershenfeld, and M. G. Kubinec. Experimental implementation of fast quantum searching. Phys. Rev. Lett., 80:3408– 3411, 1998.

[10] I. L. Chuang, N. Gershenfeld, M. G. Kubinec, and D. W. Leung. Bulk quantum computation with nuclear magnetic resonance: Theory and experiment. Proc. R. Soc. Lond. A, 454:447–467, 1998.

[11] D. G. Cory, A. F. Fahmy, and T. F. Havel. Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing. In Proceedings of the Fourth Workshop on Physics and Computation, Boston, MA, 1996. New England Complex Systems Institute.

[12] D. G. Cory, A. F. Fahmy, and T. F. Havel. Ensemble quantum computing by nuclear magnetic resonance spectroscopy. Proc. Natl. Acad. Sci., 94:1634–1639, 1997.

[13] D. G. Cory, W. Mass, M. Price, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo. Experimental quantum error correction. Phys. Rev. Lett., 81:2152–2155, 1998.

[14] D. G. Cory, M. D. Price, and T. F. Havel. Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing. Physica D, 120:82–101, 1998.

[15] C. J. L. Doran, A. N. Lasenby, and S. F. Gull. States and operators in the spacetime algebra. Found. Phys., 23:1239–1264, 1993.

[16] C. J. L. Doran, A. N. Lasenby, S. F. Gull, S. S. Somaroo, and A. D. Challinor. Spacetime algebra and electron physics. In P. Hawkes, editor, Advances in Imaging and Electron Physics, pages 271–386. Academic Press, Englewood Cliffs, NJ, 1996.

[17] R. R. Ernst, G. Bodenhausen, and A. Wokaun. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Oxford Univ. Press, U.K., 1987.

[18] N. A. Gershenfeld and I. L. Chuang. Bulk spin-resonance quantum computation. Science, 275:350–356, 1997.

[19] L. K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett., 79:325–328, 1997.

[20] D. Guilini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stametescu and H. D. Zeh. Decoherence and the Appearance of a Classical World in Quantum Theory. Springer-Verlag, Berlin, Heidelberg & New York, 1996.

[21] L. Hardy. Quantum mechanics, local realistic theories, and Lorentzinvariant realistic theories. Phys. Rev. Lett., 68:2981–2984, 1992.

[22] J. A. Jones, M. Mosca, and R. H. Hansen. Implementation of a quantum search algorithm on a quantum computer. Nature, 393:344–346, 1998.

[23] J. A. Jones and M. Mosca. Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer. J. Chem. Phys., 109:1648–1653, 1998.

[24] T. F. Jordan. Quantum mysteries explored. Am. J. Phys., 62:874–880, 1994.

[25] E. Knill, I. Chuang, and R. Laflamme. Effective pure states for bulk quantum computation. Phys. Rev. A, 57:3348–3363, 1998.

[26] E. Knill and R. Laflamme. Theory of quantum error-correcting codes. Phys. Rev. A, 55:900–911, 1997.

[27] S. Lloyd. Microscopic analogues of the Greenberger-Horne-Zeilinger experiment. Phys. Rev. A, 57:R1473–R1476, 1998.

[28] N. D. Mermin. Quantum mysteries refined. Am. J. Phys., 62:880–887, 1994.

[29] G. Navon, Y.-Q. Song, T. Rõõm, S. Appelt, R. E. Taylor, and A. Pines. Enhancement of solution NMR and MRI with laser-polarized xenon. Science, 271:1848–1851, 1996.

[30] M. A. Nielsen, E. Knill and R. Laflamme. Complete quantum teleportation using nuclear magnetic resonance. Nature, 396:52–55, 1998.

[31] J. Preskill. Unpublished lecture notes for a course on quantum compu-

tation. Available from “http://theory.caltech.edu/~preskill”.

[32] L. J. Schulman and U. Vazirani. Scalable NMR quantum computation, 1998. LANL preprint quant-ph/9804060.

[33] C. P. Slichter Principles of Magnetic Resonance (3rd ed.). Springer-Verlag, Berlin, Heidelberg & New York, 1990.

[34] S. S. Somaroo, D. G. Cory, and T. F. Havel. Expressing the operations of quantum computing in multiparticle geometric algebra. Phys. Lett. A, 240:1–7, 1998.

[35] O. W. Sörensen. Polarization transfer experiments in high-resolution NMR spectoscopy. Prog. NMR Spect., 21:503–569, 1989.

[36] O. W. Sörensen, G. W. Eich, M. H. Levitt, G. Bodenhausen, and R. R. Ernst. Product operator formalism for the description of NMR pulse experiments. Prog. NMR Spect., 16:163–192, 1983.

[37] A. M. Steane. Quantum computing. Rep. Prog. Theor. Phys., 77:793– 797, 1998.

[38] F. J. M. van de Ven and C. W. Hilbers. A simple formalism for the description of multiple-pulse experiments. Application to a weakly coupled two-spin (I = \frac{1}{2}) system. J. Magn. Reson., 54:512–520, 1983.

[39] W. S. Warren. The usefulness of NMR quantum computing. Science, 277:1688–1689, 1997. See also response by N. Gershenfeld and I. Chuang, pp. 1689–1690.

[40] C. P. Williams and S. H. Clearwater. Explorations in quantum computing. TELOS (Springer-Verlag), Santa Clara, CA, 1998.