A high-speed quantum algorithm for exhaustive searching
Lov is a researcher at Bell Labs, the research arm of Lucent Technologies. Lov can be contacted at [email protected]
When solid-state transistors were first invented in 1947, they were a few centimeters wide. Since then, the sizes of transistors have been shrinking steadily. In 1976, Gordon Moore quantified this trend by proposing that the size of transistors would keep shrinking at such a rate that the area of transistors would keep falling by a factor of 2 every eighteen months. This came to be known as "Moore's Law," and has closely predicted the evolution of transistor sizes in the last two decades.
As devices get smaller, quantum effects become more and more significant. In fact, in present day transistors, quantum mechanical effects are already important and must be considered so as not to let them affect the working of devices. Soon, it will be necessary to harness these effects. The question arises as to whether it will still be possible to design working circuits. The answer is that it will still be possible; however, the principles of design in the new regime will be very different. One way would be to redesign the circuits so that the information itself is processed quantum mechanically. Such a device would be a quantum computer. It has been shown that anything that can be computed by a classical computer could be computed equally efficiently using a quantum computer. This is significant because it shows that quantum effects do not pose an insurmountable barrier to computer design.
Quantum mechanical computers were first proposed in the early 1970s. In these, instead of being in a 0 state or a 1 state, each node can be in both states simultaneously. Therefore, it is possible to carry out multiple computations at the same time, and with proper design, it is possible to take advantage of this parallelism. Recently, two such quantum algorithms for the tasks of factorization and search have been invented. This has generated a lot of interest in quantum computers among both physicists and computer scientists.
Quantum hardware poses challenges of its own, not only because it must deal with microscopic particles that behave differently from classical objects, but also because quantum mechanical systems are very delicate; the slightest external perturbation affects the computation. The hardware must be extremely well isolated from the environment. So far, researchers have only been able to build machines with about 7 quantum bits operating at once, and even that only for a few microseconds.
From Probabilistic to Quantum Systems
Quantum mechanics is a theory about the properties of very small things. For those with an engineering background, this theory is most easily grasped as an extension of classical probability.
To describe a classical probabilistic system, you need to specify the probabilities in each state. Quantum mechanical systems have a deeper structure than classical systems. In addition to having a certain probability of being in each state, quantum mechanical systems also have a "phase" associated with the probability in each state that leads to wave-like interference as the system evolves. The complete specification of the state of the system requires the specification of both the magnitude and phase of the probability. This is usually specified as a complex number called the "amplitude" the absolute square of this complex number gives the probability of the system being in the respective state. The complete description of a quantum mechanical system is given by its amplitude vector; that is, the specification of the amplitude in each state (the amplitude vector is also known as the "wavefunction" or "superposition").
The dynamics of a quantum mechanical system is based on the amplitude vector. Just as the evolution of a probabilistic system is described by considering the initial probabilities and the transition probabilities between various states, the evolution of a quantum system is obtained by considering the initial amplitudes and the transitions between various states. In Figure 1, for instance, the initial state of the quantum mechanical system is specified by the amplitudes of an electron at the two slits. The evolution of the system (that is, the distribution of the electron at the screen) is determined by these amplitudes.
Bits and Qubits
Just as classical Boolean circuits are synthesized out of two-state systems called "bits," quantum mechanical circuits are synthesized out of two-state quantum mechanical systems called "qubits." The difference is that a bit is in either a 0 or a 1 state, whereas a qubit is in both 0 and 1 states at the same time with different amplitudes in the two states. To completely specify a classical bit requires us to specify just one bit of information 0 or 1. To specify a qubit requires us to specify the amplitudes in both states as two complex numbers. For example, (0.707 exp(i/4), 0.707 exp(-i/4)) is an amplitude vector that specifies the amplitudes in the 0 and 1 states, respectively. The probabilities in the two states are the absolute squares of these amplitudes; that is, the probabilities in each of the 0 and the 1 state is 0.5. Since the sum of the probabilities in the two states is unity, it indicates that we have a valid amplitude vector.
The analysis of quantum systems is based on the fact that the evolution of quantum systems is linear in the amplitudes. What this means is that if the amplitude vector (1,1) evolves into (2,2), and under the same physical conditions the amplitude vector (1,1) evolves into (2,2), then the amplitude vector (1+1,1+1) must evolve into (2+2,2+2). The fact that this linearity must hold for all valid values of s and s, along with reasonable physical assumptions, leads to most of the structure of quantum mechanics.
One aspect of this structure follows from the fact that any transition must conserve the overall probability. This, along with the linearity requirement, leads to the constraint that only the class of "reversible" gates can be synthesized through quantum computing. A reversible gate is one for which it is possible to deduce what the values of the inputs were from the outputs. An example of a reversible gate is NOT; an example of a gate that is not reversible is NAND.
Fortunately, any Boolean function that can be computed by standard NAND and NOR gates can also be computed by a circuit that is of roughly the same size and composed entirely of reversible gates such as NOT and XOR. Thus, any Boolean function that can be computed classically can be equally easily computed quantum mechanically (see Figures 5 and 6).
Measurement and Collapse
The characteristic of multiparticle quantum systems that leads to many of their puzzling effects is the collapse of the wavefunction. Whenever any component of the wavefunction is observed, the rest of the wavefunction immediately readjusts itself so as to be consistent with the observation.
Some of the consequences of this were so puzzling that Einstein at one time thought he had disproved quantum mechanics. However, after some thought, Bohr pointed out that the results, though paradoxical, did not violate any physical law and quantum mechanics has survived more than six decades of intense theoretical and experimental scrutiny.
As far as quantum computation is concerned, this means that it is not allowed to externally observe a system in the middle of its computation. The algorithm has to be designed in such a way that it carries out its computation completely, only after such time is the observation made. In case intermediate observations are made, their effect on the computation must be carefully analyzed (intermediate observations are not used in the quantum search algorithm).
In the mid 1930s, British mathematician Alan Turing showed that if something could be computed on a classical computer, it could be computed on a different classical computer with approximately the same amount of resources. Now, the complexity of different problems could be studied without regard to what computer they would be implemented on. This gave rise to a theory of computation. There has been tremendous work in this area in the last half-century; however, the foundations of this theory have stayed virtually unaltered.
In the early 1980s, the famous physicist Richard Feynman, in a pioneering paper, observed that there were certain problems that could easily be solved by any quantum computer, yet were very difficult for a classical computer. During the mid '80s and early '90s several such problems were studied. This showed that all the previously known results of the theory of computation were merely approximations to the real theory of computation, just as Newton's theory of gravitation had been an approximation to the real theory that was only discovered by Einstein several centuries later.
As mentioned previously, one of the constraints with quantum mechanical operations is that each operation needs to be reversible. This means that in quantum algorithms, too, this needs to be kept in mind, and each statement needs to be reversible. Assignment statements such as a=5 can be used during initialization but are not allowed within the quantum program since they irreversibly overwrite the value of a. On the other hand, statements such as a=a+5 are allowed because, if we are given the final value of a, it is possible to deduce what the initial value was.
The potential advantage of quantum computing is the built-in parallelism that allows multiple computations to be performed simultaneously. The problem is this: Even after this simultaneous computation, when all the answers are in the system, there is no easy way of extracting all this information. For a long time, it was not known whether quantum computers could solve any problem of practical interest faster than classical computers. After all, if classical computers could solve any problem of interest as fast as quantum computers, it was really not worth the effort of designing and building quantum computers. That has changed in the last five years with the discovery of fast quantum computing algorithms for factorization and searching.
The Quantum Search Algorithm
In the exhaustive search problem, a function f(x),x=0,1...(N-1), is given. The only thing we know about f(x) is that it is a binary function that is 0 everywhere except at a single value of x, where it is 1. The goal is to find the value of x for which f(x)=1. Clearly, any classical algorithm will need order N steps. However, as mentioned earlier, quantum computers can be in multiple states and carry out multiple computations at the same time if we were given a quantum mechanical circuit that evaluated f(x), there is no clear bound as to how fast the search could be done. It was shown in 1994, by using subtle properties of quantum mechanical operations, that even a quantum computer would need at least order N steps to carry out the search. Two years later in 1996, I discovered the quantum search algorithm that took precisely order N steps.
The quantum search algorithm starts out by setting the system to a superposition of N states corresponding to the N items to be searched. It can now examine all N items in a single step. However, if it is programmed to immediately print out the item examined, it will print a random item and therefore will only print out the right item with a probability of 1/N. Instead, by carrying out a set of quantum mechanical operations, it is possible to increase the amplitude and hence the probability in the desired state at the expense of other states. After this, it will indeed print out the right value of x with a high probability.
The steps required to increase the amplitude in the desired state are easiest understood by first thinking of the simple classical probabilistic program in Figure 2, which works similarly by increasing the probability with each iteration. After this, I'll address the issue of how to increase probability amplitudes with each iteration in a quantum computer; the general idea is similar.
The success probability increases by approximately 1/N in each iteration of the for loop and therefore in order N iterations, it will find the solution with a high probability.
The program as it stands above will not work quantum mechanically. The reason is that each component of a quantum program needs to be reversible. In the program in Figure 2, the statement: if(f(r)==1) answer=r; is not reversible because if you are given the value of all the variables after the statement, it may not be possible to determine what the variables were before the statement. Also, the statement: r=random(N); is not reversible because once the value of r gets overwritten, it is not possible to determine what the original value of r was. As mentioned in an earlier section, it is always possible to design a quantum mechanical algorithm that accomplishes approximately the same as any classical algorithm with roughly the same hardware. The quantum algorithm will work in terms of amplitudes that are analogous to probabilities, but can be either positive or negative.
It is possible to have a quantum random number generator, but for this to be reversible, it will need to depend on the initial state of the system. This sounds contradictory because a random number generator is supposed to generate all numbers with equal probability, irrespective of the initial state of the system. The way out of this contradiction is to let the quantum random number generator return all numbers with equal probabilities, but with the amplitudes having different signs. The information about the initial state is stored in the signs. If these signs are properly chosen, the initial state can be recovered and the random number generator can indeed be made reversible, even though each number is returned with the same probability. The following paragraph describes one such choice of signs (in quantum computing jargon, this random number generator is known as the "Walsh-Hadamard Transformation" although in this article we denote it by qrandom()). qrandom(N,r) generates a number, say q, between 0 and (N-1) with an amplitude whose magnitude is 1/N for each number. The sign of this amplitude is positive or negative and determined by the following calculation. Consider the binary representation for q and r. If the number of positions at which the binary forms of both q and r are 1 is even, the sign of the amplitude is positive; if odd, the sign is negative. For example, if q=01110101 and r=10110111, then the places where q and r both have 1s are (counting from the least significant bit): 1st, 3rd, 5th, and 6th. Therefore, the number of 1s at common positions is 4, which is even. Therefore, the sign of the amplitude will be positive. It can be shown that the operation r=qrandom(N,r) can be reversed by repeating it.
In addition to the quantum random number generator, the other quantum mechanical operation we use is a phase inversion operation. This keeps the probability in each state the same but inverts the phase of the amplitude in certain states. It is easily seen that a phase inversion operation can be reversed by applying it twice. Using these operations, it is possible to solve the search problem of size N in only about N steps. The quantum program in Figure 3 accomplishes this. It may be verified that the program is composed entirely of reversible operations.
It must be emphasized that the program of Figure 3 is inherently quantum mechanical and, as it stands, cannot be implemented on classical hardware. The fundamental difference between this and any classical probabilistic program is that this works in terms of amplitudes, which can be either positive or negative, whereas the classical program works in terms of probabilities, which are required to be positive.
Why does the quantum mechanical algorithm require only order N iterations? The analysis of the algorithm is harder than that of the classical algorithm and based on the properties of the function qrandom(). Rather than give a general analysis, the next section considers a special case. The rough idea as to why the quantum algorithm needs only order N iterations is presented in the next paragraph.
When the classical system is initialized by a randomization operation, there is a 1/N probability for each value of r as a result, each iteration of the classical algorithm increases the probability of the right answer by approximately 1/N. Similarly, when the quantum system is initialized, the probability for each value of r is 1/N; therefore the amplitude for each r, which is the square root of the probability, is 1/N. Since the quantum system works with amplitudes, each iteration of the for loop can increase the amplitude of the right answer by approximately 1/N. The phase inversion operations ensure that the amplitudes for transitions into the right answer add up while the amplitudes for transitions into other possibilities cancel out. Indeed, in order N iterations of the for loop, the amplitude of the right answer rises to a number of order 1. The probability being the square of the amplitude also rises to order 1.
The N=4 Case
In general, eta (the number of iterations required by the quantum search algorithm), is a number between 0.5N and 0.8N. Its precise value can be analytically obtained. The precise value is important for the correct working of the algorithm because, unlike in the classical case, the success probability is not monotonic with the number of iterations. The success probability oscillates between 0 and 1 with a certain periodicity. If we stop exactly at the right number of iterations, the success probability is close to 1. For the N=4 case, eta is exactly 1 and the for loop in the algorithm needs to be executed just once. Figure 4 shows the steps of this program.
As mentioned in the previous section, qrandom(N,r) generates a random number between 0 and (N-1) with an amplitude of 1/N. In particular, qrandom(4,r) returns a number (q) in the range [0,3] with amplitude 0.5. The sign depends on r and q, as shown in Figure 7. In Figure 8, the sign of the amplitudes is calculated, as described in the previous section, by counting the number of positions at which the binary forms of both q and r are 1. For example, consider the case when q and r are both 1 (encircled in Figure 8). The binary form of q (01) and the binary form of r (01) both have a 1 in the 2nd position. Since the number of 1s in the same positions are odd, the sign of the amplitude is negative and the amplitude of qrandom(4,1) returning a 1 is -0.5.
With the information in Figure 8, it is a straightforward, though tedious, exercise to analyze quantum_main(). The analysis, as shown in Figure 8, consists of calculating the amplitude vectors for the system (in this case, r) at each step of the program just as we would list out the value of all variables at each step when debugging a classical program. The difference is that a classical variable assumes only one value and you just need to list that one value. On the other hand, a quantum system is in all states with certain amplitudes and we need to list out the amplitudes for all possible states.
As in Figure 8, after carrying out the steps of the quantum_main() program, the amplitude of the correct number is -1 (the probability is, hence, 1) and the amplitudes of all other numbers is 0 (the probabilities are, hence, 0). An observation now reveals the right answer with certainty. A single evaluation of f(x) (in Step 2), along with some preprocessing and postprocessing operations, suffices to identify the answer with certainty in the N=4 case. In the classical case, it would take three evaluations of f(x) to locate the correct number. A similar analysis for the general case shows that it can identify the answer in only order N steps. Also, the quantum algorithm is able to cancel the amplitudes in all the incorrect possibilities since certain transition amplitudes are negative. Hence, it can find the correct answer with certainty. A classical probabilistic algorithm, like the one described in the previous section, will have a finite error probability. The following is an outline of the calculations of the amplitude vector for r at each step of the program in Figure 4:
1. The first step is qrandom(4,0). According to Figure 7, this creates an amplitude of +0.5 for each of the four values of r. The amplitude vector becomes (0.5, 0.5, 0.5, 0.5).
2. Next, the amplitude is inverted for the value of r satisfying f(r)=1. Assuming this is the third number, the amplitude vector becomes (0.5, 0.5, -0.5, 0.5). This is the only step that requires an evaluation of f(x).
3. After this, the operation r=qrandom(4,r) is carried out, which seems to randomize r. However, as mentioned earlier, the transition amplitudes have carefully chosen signs that preserve the information. After Step 3, r can assume any value in the range [0-3] with certain amplitudes that need to be calculated in a way very similar to those for probabilities. To illustrate this, the amplitude for r to be 0, after Step 3, is calculated.
There are four paths through which r can reach 0 after Step 3. To estimate this amplitude, we need to sum the amplitudes over these four paths. The four paths are:
(a) r was 0 before Step 3 and stays unchanged during Step 3.
(b) r was 1 before Step 3 and Step 3 causes it to make a transition from 1 to 0.
(c) r was 2 before Step 3 and Step 3 causes it to make a transition from 2 to 0.
(d) r was 3 before Step 3 and Step 3 causes it to make a transition from 3 to 0.
Just as for probabilities, we calculate the amplitudes of each of these four paths and sum them to find the total amplitude. The amplitude of (a) is given by multiplying the amplitude of r being initially 0 (0.5) by the amplitude of it staying 0 during Step 3 (this is read off from Figure 7 and is 0.5). Thus, the amplitude of a is 0.5×0.5, which is 0.25. Similarly, the amplitude of (b) is 0.25, that of (c) is -0.25, and that of (d) is 0.25. Summing these four amplitudes leads to a total amplitude of 0.5 for r to be 0 after Step 3. Carrying out these calculations for the other 3 possible values of r leads to the result that the amplitude vector for r after Step 3 is (0.5, -0.5, 0.5, 0.5).
4. Next the amplitude in the 0 state is inverted and the amplitude vector for r becomes: (-0.5, -0.5, 0.5, 0.5).
5. Finally the operation r=qrandom(4,r); is applied again and as in Step 3, the amplitudes of r having different values can be calculated by considering the paths through which r can reach each value. After Step 5, the amplitude vector for r can be shown to be: (0.0, 0.0, -1.0, 0.0).
Quantum mechanical systems are very delicate and have to be designed to be almost completely isolated from their environments. This is because, as mentioned previously, whenever any portion of a quantum system is observed, the rest of the system readjusts itself so as to become consistent with the observation. Thus any time a quantum system interacts with its environment (any macroscopic system, for instance), it gets perturbed just as if someone had tried to observe it. This affects the computation in unpredictable ways. Isolating a large system is difficult. Our environment contains a lot of background noise, such as stray dust particles and stray photons that rapidly perturb any macroscopic body. For example, a football lasts less than 10-20 seconds before it is struck by a stray particle. On the other hand, tiny particles can stay isolated for much longer an atom can stay isolated for hours or even days. The problem is that it is much harder to carry out controllable operations on a quantum mechanical system of atomic scale. Some techniques through which elementary quantum computing operations have been demonstrated are:
- Single photons interacting through nonlinear media.
- Ions in a cryogenic trap.
- Nuclear magnetic resonance (NMR).
- Superconducting devices in solids.
NMR is the most advanced, having demonstrated 7-qubit systems. Simple algorithms including versions of the quantum search algorithm have been implemented using NMR in organic liquids.
Considering the experience of integrated circuits, in the long term, solids appear the most promising. However, there are several problems with solids; for example, to achieve isolation in solids, we need to go to very low temperatures.
Quantum computing is a field in its infancy. The two challenges in the field are the hardware and the software. The hardware challenge is to devise schemes for building quantum hardware that can naturally be isolated from the environment and on which controllable quantum operations can be carried out. The problem is that such a device will be radically different from any known device and no one knows what its structure will be like. So far, only very small prototypes have been built. The software challenge is to devise applications that will make the task of building quantum hardware worthwhile. Currently, there are just two important applications known search and factorization.
When will a quantum computer be built? It is interesting to recall the following quote from a visionary Popular Mechanics article of 1949 "Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs 30 tons, computers in the future may have only 1000 vacuum tubes and weigh only 1.5 tons."
Gershenfeld, Neil and Isaac Chuang. "Quantum Computing With Molecules," Scientific American, June 1998.
Preskill, John. "Quantum Computing: Pro & Con," http://xxx.lanl.gov/abs/quant-ph/9705032.
Grover, Lov K. "From Schrodinger's Equation to the Quantum Search Algorithm," http://www.bell-labs.com/user/lkgrover/papers.html.
Bennett, Charles H. and David P. DiVincenzo. "Quantum Information And Computation," Nature, March 16, 2000.