Quantum search algorithm
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, refers to a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just O ( N ) {displaystyle O({sqrt {N}})} evaluations of the function, where N {displaystyle N} is the size of the function's domain. It was devised by Lov Grover in 1996.[1]
The analogous problem in classical computation cannot be solved in fewer than O ( N ) {displaystyle O(N)} evaluations (because, on average, one has to check half of the domain to get a 50% chance of finding the right input). Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function ( N ) {displaystyle Omega ({sqrt {N}})} times, so Grover's algorithm is asymptotically optimal.[2] Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests that Grover's algorithm by itself will not provide polynomial-time solutions for NP-complete problems (as the square root of an exponential function is an exponential, not polynomial, function).[3]
Unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts, Grover's algorithm provides only a quadratic speedup. However, even quadratic speedup is considerable when N {displaystyle N} is large, and Grover's algorithm can be applied to speed up broad classes of algorithms.[3] Grover's algorithm could brute-force a 128-bit symmetric cryptographic key in roughly 264 iterations, or a 256-bit key in roughly 2128 iterations. As a result, it is sometimes suggested[4] that symmetric key lengths be doubled to protect against future quantum attacks.
Grover's algorithm, along with variants like amplitude amplification, can be used to speed up a broad range of algorithms.[5][6][7] In particular, algorithms for NP-complete problems generally contain exhaustive search as a subroutine, which can be sped up by Grover's algorithm.[6] The current best algorithm for 3SAT is one such example. Generic constraint satisfaction problems also see quadratic speedups with Grover.[8] These algorithms do not require that the input be given in the form of an oracle, since Grover's algorithm is being applied with an explicit function, e.g. the function checking that a set of bits satisfies a 3SAT instance.
Grover's algorithm can also give provable speedups for black-box problems in quantum query complexity, including element distinctness[9] and the collision problem[10] (solved with the BrassardHyerTapp algorithm). In these types of problems, one treats the oracle function f as a database, and the goal is to use the quantum query to this function as few times as possible.
Grover's algorithm essentially solves the task of function inversion. Roughly speaking, if we have a function y = f ( x ) {displaystyle y=f(x)} that can be evaluated on a quantum computer, Grover's algorithm allows us to calculate x {displaystyle x} when given y {displaystyle y} . Consequently, Grover's algorithm gives broad asymptotic speed-ups to many kinds of brute-force attacks on symmetric-key cryptography, including collision attacks and pre-image attacks.[11] However, this may not necessarily be the most efficient algorithm since, for example, the parallel rho algorithm is able to find a collision in SHA2 more efficiently than Grover's algorithm.[12]
Grover's original paper described the algorithm as a database search algorithm, and this description is still common. The database in this analogy is a table of all of the function's outputs, indexed by the corresponding input. However, this database is not represented explicitly. Instead, an oracle is invoked to evaluate an item by its index. Reading a full database item by item and converting it into such a representation may take a lot longer than Grover's search. To account for such effects, Grover's algorithm can be viewed as solving an equation or satisfying a constraint. In such applications, the oracle is a way to check the constraint and is not related to the search algorithm. This separation usually prevents algorithmic optimizations, whereas conventional search algorithms often rely on such optimizations and avoid exhaustive search.[13]
The major barrier to instantiating a speedup from Grover's algorithm is that the quadratic speedup achieved is too modest to overcome the large overhead of near-term quantum computers.[14] However, later generations of fault-tolerant quantum computers with better hardware performance may be able to realize these speedups for practical instances of data.
As input for Grover's algorithm, suppose we have a function f : { 0 , 1 , , N 1 } { 0 , 1 } {displaystyle fcolon {0,1,ldots ,N-1}to {0,1}} . In the "unstructured database" analogy, the domain represent indices to a database, and f(x) = 1 if and only if the data that x points to satisfies the search criterion. We additionally assume that only one index satisfies f(x) = 1, and we call this index . Our goal is to identify .
We can access f with a subroutine (sometimes called an oracle) in the form of a unitary operator U that acts as follows:
This uses the N {displaystyle N} -dimensional state space H {displaystyle {mathcal {H}}} , which is supplied by a register with n = log 2 N {displaystyle n=lceil log _{2}Nrceil } qubits.This is often written as
Grover's algorithm outputs with probability at least 1/2 using O ( N ) {displaystyle O({sqrt {N}})} applications of U. This probability can be made arbitrarily large by running Grover's algorithm multiple times. If one runs Grover's algorithm until is found, the expected number of applications is still O ( N ) {displaystyle O({sqrt {N}})} , since it will only be run twice on average.
This section compares the above oracle U {displaystyle U_{omega }} with an oracle U f {displaystyle U_{f}} .
U is different from the standard quantum oracle for a function f. This standard oracle, denoted here as Uf, uses an ancillary qubit system. The operation then represents an inversion (NOT gate) on the main system conditioned by the value of f(x) from the ancillary system:
or briefly,
These oracles are typically realized using uncomputation.
If we are given Uf as our oracle, then we can also implement U, since U is Uf when the ancillary qubit is in the state | = 1 2 ( | 0 | 1 ) = H | 1 {displaystyle |-rangle ={frac {1}{sqrt {2}}}{big (}|0rangle -|1rangle {big )}=H|1rangle } :
So, Grover's algorithm can be run regardless of which oracle is given.[3] If Uf is given, then we must maintain an additional qubit in the state | {displaystyle |-rangle } and apply Uf in place of U.
The steps of Grover's algorithm are given as follows:
For the correctly chosen value of r {displaystyle r} , the output will be | {displaystyle |omega rangle } with probability approaching 1 for N 1. Analysis shows that this eventual value for r ( N ) {displaystyle r(N)} satisfies r ( N ) 4 N {displaystyle r(N)leq {Big lceil }{frac {pi }{4}}{sqrt {N}}{Big rceil }} .
Implementing the steps for this algorithm can be done using a number of gates linear in the number of qubits.[3] Thus, the gate complexity of this algorithm is O ( log ( N ) r ( N ) ) {displaystyle O(log(N)r(N))} , or O ( log ( N ) ) {displaystyle O(log(N))} per iteration.
There is a geometric interpretation of Grover's algorithm, following from the observation that the quantum state of Grover's algorithm stays in a two-dimensional subspace after each step. Consider the plane spanned by | s {displaystyle |srangle } and | {displaystyle |omega rangle } ; equivalently, the plane spanned by | {displaystyle |omega rangle } and the perpendicular ket | s = 1 N 1 x | x {displaystyle textstyle |s'rangle ={frac {1}{sqrt {N-1}}}sum _{xneq omega }|xrangle } .
Grover's algorithm begins with the initial ket | s {displaystyle |srangle } , which lies in the subspace. The operator U {displaystyle U_{omega }} is a reflection at the hyperplane orthogonal to | {displaystyle |omega rangle } for vectors in the plane spanned by | s {displaystyle |s'rangle } and | {displaystyle |omega rangle } , i.e. it acts as a reflection across | s {displaystyle |s'rangle } . This can be seen by writing U {displaystyle U_{omega }} in the form of a Householder reflection:
The operator U s = 2 | s s | I {displaystyle U_{s}=2|srangle langle s|-I} is a reflection through | s {displaystyle |srangle } . Both operators U s {displaystyle U_{s}} and U {displaystyle U_{omega }} take states in the plane spanned by | s {displaystyle |s'rangle } and | {displaystyle |omega rangle } to states in the plane. Therefore, Grover's algorithm stays in this plane for the entire algorithm.
It is straightforward to check that the operator U s U {displaystyle U_{s}U_{omega }} of each Grover iteration step rotates the state vector by an angle of = 2 arcsin 1 N {displaystyle theta =2arcsin {tfrac {1}{sqrt {N}}}} .So, with enough iterations, one can rotate from the initial state | s {displaystyle |srangle } to the desired output state | {displaystyle |omega rangle } . The initial ket is close to the state orthogonal to | {displaystyle |omega rangle } :
In geometric terms, the angle / 2 {displaystyle theta /2} between | s {displaystyle |srangle } and | s {displaystyle |s'rangle } is given by
We need to stop when the state vector passes close to | {displaystyle |omega rangle } ; after this, subsequent iterations rotate the state vector away from | {displaystyle |omega rangle } , reducing the probability of obtaining the correct answer. The exact probability of measuring the correct answer is
where r is the (integer) number of Grover iterations. The earliest time that we get a near-optimal measurement is therefore r N / 4 {displaystyle rapprox pi {sqrt {N}}/4} .
To complete the algebraic analysis, we need to find out what happens when we repeatedly apply U s U {displaystyle U_{s}U_{omega }} . A natural way to do this is by eigenvalue analysis of a matrix. Notice that during the entire computation, the state of the algorithm is a linear combination of s {displaystyle s} and {displaystyle omega } . We can write the action of U s {displaystyle U_{s}} and U {displaystyle U_{omega }} in the space spanned by { | s , | } {displaystyle {|srangle ,|omega rangle }} as:
So in the basis { | , | s } {displaystyle {|omega rangle ,|srangle }} (which is neither orthogonal nor a basis of the whole space) the action U s U {displaystyle U_{s}U_{omega }} of applying U {displaystyle U_{omega }} followed by U s {displaystyle U_{s}} is given by the matrix
This matrix happens to have a very convenient Jordan form. If we define t = arcsin ( 1 / N ) {displaystyle t=arcsin(1/{sqrt {N}})} , it is
It follows that r-th power of the matrix (corresponding to r iterations) is
Using this form, we can use trigonometric identities to compute the probability of observing after r iterations mentioned in the previous section,
Alternatively, one might reasonably imagine that a near-optimal time to distinguish would be when the angles 2rt and 2rt are as far apart as possible, which corresponds to 2 r t / 2 {displaystyle 2rtapprox pi /2} , or r = / 4 t = / 4 arcsin ( 1 / N ) N / 4 {displaystyle r=pi /4t=pi /4arcsin(1/{sqrt {N}})approx pi {sqrt {N}}/4} . Then the system is in state
A short calculation now shows that the observation yields the correct answer with error O ( 1 N ) {displaystyle Oleft({frac {1}{N}}right)} .
If, instead of 1 matching entry, there are k matching entries, the same algorithm works, but the number of iterations must be 4 ( N k ) 1 / 2 {textstyle {frac {pi }{4}}{left({frac {N}{k}}right)^{1/2}}} instead of 4 N 1 / 2 . {textstyle {frac {pi }{4}}{N^{1/2}}.}
There are several ways to handle the case if k is unknown.[15] A simple solution performs optimally up to a constant factor: run Grover's algorithm repeatedly for increasingly small values of k, e.g., taking k = N, N/2, N/4, ..., and so on, taking k = N / 2 t {displaystyle k=N/2^{t}} for iteration t until a matching entry is found.
With sufficiently high probability, a marked entry will be found by iteration t = log 2 ( N / k ) + c {displaystyle t=log _{2}(N/k)+c} for some constant c. Thus, the total number of iterations taken is at most
A version of this algorithm is used in order to solve the collision problem.[16][17]
A modification of Grover's algorithm called quantum partial search was described by Grover and Radhakrishnan in 2004.[18] In partial search, one is not interested in finding the exact address of the target item, only the first few digits of the address. Equivalently, we can think of "chunking" the search space into blocks, and then asking "in which block is the target item?". In many applications, such a search yields enough information if the target address contains the information wanted. For instance, to use the example given by L. K. Grover, if one has a list of students organized by class rank, we may only be interested in whether a student is in the lower 25%, 2550%, 5075% or 75100% percentile.
To describe partial search, we consider a database separated into K {displaystyle K} blocks, each of size b = N / K {displaystyle b=N/K} . The partial search problem is easier. Consider the approach we would take classically we pick one block at random, and then perform a normal search through the rest of the blocks (in set theory language, the complement). If we don't find the target, then we know it's in the block we didn't search. The average number of iterations drops from N / 2 {displaystyle N/2} to ( N b ) / 2 {displaystyle (N-b)/2} .
Grover's algorithm requires 4 N {textstyle {frac {pi }{4}}{sqrt {N}}} iterations. Partial search will be faster by a numerical factor that depends on the number of blocks K {displaystyle K} . Partial search uses n 1 {displaystyle n_{1}} global iterations and n 2 {displaystyle n_{2}} local iterations. The global Grover operator is designated G 1 {displaystyle G_{1}} and the local Grover operator is designated G 2 {displaystyle G_{2}} .
The global Grover operator acts on the blocks. Essentially, it is given as follows:
The optimal values of j 1 {displaystyle j_{1}} and j 2 {displaystyle j_{2}} are discussed in the paper by Grover and Radhakrishnan. One might also wonder what happens if one applies successive partial searches at different levels of "resolution". This idea was studied in detail by Vladimir Korepin and Xu, who called it binary quantum search. They proved that it is not in fact any faster than performing a single partial search.
Grover's algorithm is optimal up to sub-constant factors. That is, any algorithm that accesses the database only by using the operator U must apply U at least a 1 o ( 1 ) {displaystyle 1-o(1)} fraction as many times as Grover's algorithm.[19] The extension of Grover's algorithm to k matching entries, (N/k)1/2/4, is also optimal.[16] This result is important in understanding the limits of quantum computation.
If the Grover's search problem was solvable with logc N applications of U, that would imply that NP is contained in BQP, by transforming problems in NP into Grover-type search problems. The optimality of Grover's algorithm suggests that quantum computers cannot solve NP-Complete problems in polynomial time, and thus NP is not contained in BQP.
It has been shown that a class of non-local hidden variable quantum computers could implement a search of an N {displaystyle N} -item database in at most O ( N 3 ) {displaystyle O({sqrt[{3}]{N}})} steps. This is faster than the O ( N ) {displaystyle O({sqrt {N}})} steps taken by Grover's algorithm.[20]
Read the original post:
Grover's algorithm - Wikipedia
- Time Crystals Could be the Key to the First Quantum Computer - TrendinTech [Last Updated On: May 3rd, 2017] [Originally Added On: May 3rd, 2017]
- The Quantum Computer Revolution Is Closer Than You May Think - National Review [Last Updated On: May 3rd, 2017] [Originally Added On: May 3rd, 2017]
- Chinese scientists build world's first quantum computing machine - India Today [Last Updated On: May 3rd, 2017] [Originally Added On: May 3rd, 2017]
- Quantum Computing | D-Wave Systems [Last Updated On: May 3rd, 2017] [Originally Added On: May 3rd, 2017]
- Quantum computing utilizes 3D crystals - Johns Hopkins News-Letter [Last Updated On: May 4th, 2017] [Originally Added On: May 4th, 2017]
- Quantum Computing and What All Good IT Managers Should Know - TrendinTech [Last Updated On: May 4th, 2017] [Originally Added On: May 4th, 2017]
- World's First Quantum Computer Made By China 24000 Times Faster Than International Counterparts - Fossbytes [Last Updated On: May 4th, 2017] [Originally Added On: May 4th, 2017]
- China adds a quantum computer to high-performance computing arsenal - PCWorld [Last Updated On: May 6th, 2017] [Originally Added On: May 6th, 2017]
- Quantum computing: A simple introduction - Explain that Stuff [Last Updated On: May 6th, 2017] [Originally Added On: May 6th, 2017]
- What is Quantum Computing? Webopedia Definition [Last Updated On: May 6th, 2017] [Originally Added On: May 6th, 2017]
- Quantum Computing Market Forecast 2017-2022 | Market ... [Last Updated On: May 6th, 2017] [Originally Added On: May 6th, 2017]
- China hits milestone in developing quantum computer - South China Morning Post [Last Updated On: May 8th, 2017] [Originally Added On: May 8th, 2017]
- China builds five qubit quantum computer sampling and will scale to 20 qubits by end of this year and could any beat ... - Next Big Future [Last Updated On: May 8th, 2017] [Originally Added On: May 8th, 2017]
- Five Ways Quantum Computing Will Change the Way We Think ... - PR Newswire (press release) [Last Updated On: May 8th, 2017] [Originally Added On: May 8th, 2017]
- Quantum Computing Demands a Whole New Kind of Programmer - Singularity Hub [Last Updated On: May 9th, 2017] [Originally Added On: May 9th, 2017]
- New materials bring quantum computing closer to reality - Phys.org - Phys.Org [Last Updated On: May 9th, 2017] [Originally Added On: May 9th, 2017]
- Researchers Invent Nanoscale 'Refrigerator' for Quantum ... - Sci-News.com [Last Updated On: May 11th, 2017] [Originally Added On: May 11th, 2017]
- China's New Type of Quantum Computing Device, Built Inside a Diamond - TrendinTech [Last Updated On: May 11th, 2017] [Originally Added On: May 11th, 2017]
- Molecular magnets closer to application in quantum computing - Next Big Future [Last Updated On: May 11th, 2017] [Originally Added On: May 11th, 2017]
- New Materials Could Make Quantum Computers More Practical - Tom's Hardware [Last Updated On: May 11th, 2017] [Originally Added On: May 11th, 2017]
- Home News Computer Europe Takes Quantum Computing to the Next Level With this Billion Euro... - TrendinTech [Last Updated On: May 13th, 2017] [Originally Added On: May 13th, 2017]
- Researchers seek to advance quantum computing - The Stanford Daily [Last Updated On: May 13th, 2017] [Originally Added On: May 13th, 2017]
- quantum computing - WIRED UK [Last Updated On: May 13th, 2017] [Originally Added On: May 13th, 2017]
- Scientists Invent Nanoscale Refrigerator For Quantum Computers - Wall Street Pit [Last Updated On: May 14th, 2017] [Originally Added On: May 14th, 2017]
- D-Wave Closes $50M Facility to Fund Next Generation of Quantum Computers - Marketwired (press release) [Last Updated On: May 17th, 2017] [Originally Added On: May 17th, 2017]
- Quantum Computers Sound Great, But Who's Going to Program Them? - TrendinTech [Last Updated On: May 17th, 2017] [Originally Added On: May 17th, 2017]
- Quantum Computing Could Use Graphene To Create Stable Qubits - International Business Times [Last Updated On: May 18th, 2017] [Originally Added On: May 18th, 2017]
- Bigger is better: Quantum volume expresses computer's limit - Ars Technica [Last Updated On: May 18th, 2017] [Originally Added On: May 18th, 2017]
- IBM's Newest Quantum Computing Processors Have Triple the Qubits of Their Last - Futurism [Last Updated On: May 18th, 2017] [Originally Added On: May 18th, 2017]
- It's time to decide how quantum computing will help your business - Techworld Australia [Last Updated On: May 20th, 2017] [Originally Added On: May 20th, 2017]
- IBM makes a leap in quantum computing power - PCWorld [Last Updated On: May 20th, 2017] [Originally Added On: May 20th, 2017]
- IBM scientists demonstrate ballistic nanowire connections, a potential future key component for quantum computing - Phys.Org [Last Updated On: May 20th, 2017] [Originally Added On: May 20th, 2017]
- The route to high-speed quantum computing is paved with error - Ars Technica UK [Last Updated On: May 20th, 2017] [Originally Added On: May 20th, 2017]
- IBM makes leap in quantum computing power - ITworld [Last Updated On: May 22nd, 2017] [Originally Added On: May 22nd, 2017]
- Researchers push forward quantum computing research - The ... - Economic Times [Last Updated On: May 22nd, 2017] [Originally Added On: May 22nd, 2017]
- Quantum Computing Research Given a Boost by Stanford Team - News18 [Last Updated On: May 22nd, 2017] [Originally Added On: May 22nd, 2017]
- US playing catch-up in quantum computing - The Register-Guard [Last Updated On: May 22nd, 2017] [Originally Added On: May 22nd, 2017]
- Stanford researchers push forward quantum computing research ... - The Indian Express [Last Updated On: May 23rd, 2017] [Originally Added On: May 23rd, 2017]
- NASA Scientist Eleanor Rieffel to give a talk on quantum computing - Chapman University: Happenings (blog) [Last Updated On: May 23rd, 2017] [Originally Added On: May 23rd, 2017]
- Graphene Just Brought Us One Step Closer to Practical Quantum Computers - Futurism [Last Updated On: May 23rd, 2017] [Originally Added On: May 23rd, 2017]
- IBM Q Offers Quantum Computing as a Service - The Merkle [Last Updated On: May 23rd, 2017] [Originally Added On: May 23rd, 2017]
- How quantum computing increases cybersecurity risks | Network ... - Network World [Last Updated On: May 23rd, 2017] [Originally Added On: May 23rd, 2017]
- Quantum Computing Is Going Commercial With the Potential ... [Last Updated On: May 23rd, 2017] [Originally Added On: May 23rd, 2017]
- Is the US falling behind in the race for quantum computing? - AroundtheO [Last Updated On: May 26th, 2017] [Originally Added On: May 26th, 2017]
- Quantum computing, election pledges and a thief who made science history - Nature.com [Last Updated On: May 26th, 2017] [Originally Added On: May 26th, 2017]
- Top 5: Things to know about quantum computers - TechRepublic [Last Updated On: May 26th, 2017] [Originally Added On: May 26th, 2017]
- Google Plans to Demonstrate the Supremacy of Quantum ... - IEEE Spectrum [Last Updated On: May 26th, 2017] [Originally Added On: May 26th, 2017]
- Quantum Computing Is Real, and D-Wave Just Open ... - WIRED [Last Updated On: May 26th, 2017] [Originally Added On: May 26th, 2017]
- IBM to Sell Use of Its New 17-Qubit Quantum Computer over the Cloud - All About Circuits [Last Updated On: May 28th, 2017] [Originally Added On: May 28th, 2017]
- Doped Diamonds Push Practical Quantum Computing Closer to Reality - Motherboard [Last Updated On: May 28th, 2017] [Originally Added On: May 28th, 2017]
- For more advanced computing, technology needs to make a ... - CIO Dive [Last Updated On: May 30th, 2017] [Originally Added On: May 30th, 2017]
- Microsoft, Purdue Extend Quantum Computing Partnership To Create More Stable Qubits - Tom's Hardware [Last Updated On: May 30th, 2017] [Originally Added On: May 30th, 2017]
- AI and Quantum Computers Are Our Best Weapons Against Cyber Criminals - Futurism [Last Updated On: May 30th, 2017] [Originally Added On: May 30th, 2017]
- Toward mass-producible quantum computers | MIT News - MIT News [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- Purdue, Microsoft Partner On Quantum Computing Research | WBAA - WBAA [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- Tektronix AWG Pulls Test into Era of Quantum Computing - Electronic Design [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- Telstra just wants a quantum computer to offer as-a-service - ZDNet [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- D-Wave partners with U of T to move quantum computing along - Financial Post [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- MIT Just Unveiled A Technique to Mass Produce Quantum Computers - Futurism [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- Here's how we can achieve mass-produced quantum computers ... - ScienceAlert [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- Research collaborative pursues advanced quantum computing - Phys.Org [Last Updated On: June 1st, 2017] [Originally Added On: June 1st, 2017]
- Team develops first blockchain that can't be hacked by quantum computer - Siliconrepublic.com [Last Updated On: June 3rd, 2017] [Originally Added On: June 3rd, 2017]
- Quantum computers to drive customer insights, says CBA CIO - CIO - CIO Australia [Last Updated On: June 6th, 2017] [Originally Added On: June 6th, 2017]
- FinDEVr London: Preparing for the Dark Side of Quantum Computing - GlobeNewswire (press release) [Last Updated On: June 8th, 2017] [Originally Added On: June 8th, 2017]
- Scientists May Have Found a Way to Combat Quantum Computer Blockchain Hacking - Futurism [Last Updated On: June 9th, 2017] [Originally Added On: June 9th, 2017]
- Purdue, Microsoft to Collaborate on Quantum Computer - Photonics.com [Last Updated On: June 9th, 2017] [Originally Added On: June 9th, 2017]
- From the Abacus to Supercomputers to Quantum Computers - Duke Today [Last Updated On: June 12th, 2017] [Originally Added On: June 12th, 2017]
- Microsoft and Purdue work on scalable topological quantum computer - Next Big Future [Last Updated On: June 12th, 2017] [Originally Added On: June 12th, 2017]
- Are Enterprises Ready to Take a Quantum Leap? - IT Business Edge [Last Updated On: June 12th, 2017] [Originally Added On: June 12th, 2017]
- A Hybrid of Quantum Computing and Machine Learning Is Spawning New Ventures - IEEE Spectrum [Last Updated On: June 14th, 2017] [Originally Added On: June 14th, 2017]
- The Machine of Tomorrow Today: Quantum Computing on the Verge - Bloomberg [Last Updated On: June 14th, 2017] [Originally Added On: June 14th, 2017]
- KPN CISO details Quantum computing attack dangers - Mobile World Live [Last Updated On: June 15th, 2017] [Originally Added On: June 15th, 2017]
- Accenture, Biogen, 1QBit Launch Quantum Computing App to ... - HIT Consultant [Last Updated On: June 15th, 2017] [Originally Added On: June 15th, 2017]
- Angry Birds, qubits and big ideas: Quantum computing is tantalisingly close - The Australian Financial Review [Last Updated On: June 15th, 2017] [Originally Added On: June 15th, 2017]
- Consortium Applies Quantum Computing to Drug Discovery for Neurological Diseases - Drug Discovery & Development [Last Updated On: June 15th, 2017] [Originally Added On: June 15th, 2017]
- Accenture, 1QBit partner for drug discovery through quantum computing - ZDNet [Last Updated On: June 15th, 2017] [Originally Added On: June 15th, 2017]
- How to get ahead in quantum machine learning AND attract Goldman Sachs - eFinancialCareers [Last Updated On: June 15th, 2017] [Originally Added On: June 15th, 2017]
- Quantum computing, the machines of tomorrow - The Japan Times [Last Updated On: June 16th, 2017] [Originally Added On: June 16th, 2017]
- Toward optical quantum computing - MIT News [Last Updated On: June 17th, 2017] [Originally Added On: June 17th, 2017]
- Its time to decide how quantum computing will help your ... [Last Updated On: June 18th, 2017] [Originally Added On: June 18th, 2017]