A New Complexity Theory for the Quantum Age

Traditional complexity theory can’t accommodate problems with quantum inputs and outputs.

传统的复杂性理论无法解决量子输入和输出的问题。

Henry Yuen wants to build a new theory that can.

亨利·袁（Henry Yuen）想要建立一个可以做到的新理论。

Computer science, at its most fundamental, is all about inputs and outputs.

计算机科学从根本上来说就是关于输入和输出。

Consider the simple case of multiplying two numbers on a pocket calculator.

考虑在袖珍计算器上将两个数字相乘的简单情况。

You punch in some inputs — the specific numbers you want to multiply — and the screen displays an output representing their product.

你输入一些输入——你想要相乘的具体数字——屏幕就会显示代表它们乘积的输出。

The reverse problem of breaking a number into its prime factors can be much harder, but it has the same basic structure.

将数字分解为其质因数的逆向问题可能要困难得多，但它具有相同的基本结构。

Solving a problem on a computer always boils down to transforming numerical inputs, usually written as strings of 0s and 1s, into outputs.

在计算机上解决问题总是归结为将数字输入（通常写为 0 和 1 的字符串）转换为输出。

Researchers in the field of computational complexity theory study why some of these transformations are harder to implement than others.

计算复杂性理论领域的研究人员研究了为什么其中一些转换比其他转换更难实现。

They’ve discovered that certain tasks that ordinary “classical” computers struggle with, like the prime factor problem, are much easier for computers that exploit the laws of quantum physics.

他们发现，普通“经典”计算机难以完成的某些任务，例如素因数问题，对于利用量子物理定律的计算机来说要容易得多。

For over 30 years, complexity theorists have used this theoretical framework to identify problems where quantum computers surpass classical ones.

30 多年来，复杂性理论学家一直使用这个理论框架来识别量子计算机超越经典计算机的问题。

But there’s a broader class of problems that they’ve barely begun to study, whose inputs and outputs aren’t ordinary strings of bits.

但还有一类更广泛的问题他们刚刚开始研究，其输入和输出不是普通的比特串。

These are the problems that most interest the complexity theorist Henry Yuen .

这些是复杂性理论家亨利·袁最感兴趣的问题。

How do you make sense of problems whose inputs and outputs are themselves inherently quantum?

您如何理解输入和输出本身就是量子的问题？

“Traditional complexity theory is just silent on this,” Yuen said.

“传统的复杂性理论对此只字不提，”袁说。

“Maybe we need a separate theory to understand this other class of problems.”

“也许我们需要一个单独的理论来理解另一类问题。”

Yuen, a professor at Columbia University who co-authored a landmark proof in traditional complexity theory in 2020, is now leading an ambitious effort to build a new “fully quantum” theory that can accommodate these unusual inputs and outputs.

哥伦比亚大学教授 Yuen 在 2020 年与人合着了传统复杂性理论中具有里程碑意义的证明，目前正在领导一项雄心勃勃的努力，以建立一种新的“全量子”理论，以适应这些不寻常的输入和输出。

His own life speaks to what’s possible with atypical inputs.

他自己的生活说明了非典型投入的可能性。

Born in 1989, he grew up working in his family’s Southern California restaurant, the son of refugees from rural Cambodia who had fled genocide in the 1970s.

他出生于 1989 年，在自家位于南加州的餐厅工作长大，父亲是逃离 20 世纪 70 年代种族灭绝的柬埔寨农村难民的儿子。

He learned computer programming because he wanted to design video games.

他学习计算机编程是因为他想设计视频游戏。

That early interest led him to major in computer science in college, where he grew fascinated by the theoretical foundations of quantum computing.

这种早期的兴趣促使他在大学主修计算机科学，在那里他对量子计算的理论基础着迷。

Quanta spoke with Yuen about where complexity theory falls short, what quantum cryptography has to do with black holes, and the importance of open-ended research.

广达与袁讨论了复杂性理论的不足之处、量子密码学与黑洞的关系以及开放式研究的重要性。

The interview has been condensed and edited for clarity.

为了清晰起见，采访内容已经过精简和编辑。

In traditional complexity theory, the inputs and outputs are always classical.

在传统的复杂性理论中，输入和输出总是经典的。

What happens in between is up to you.

中间发生什么取决于你。

You can try to use a classical computer to solve your problem, or you can use a quantum computer, and we’re hoping that quantum computers will be better at some problems.

你可以尝试使用经典计算机来解决你的问题，也可以使用量子计算机，我们希望量子计算机在某些问题上能做得更好。

But why do inputs and outputs have to be classical?

但为什么输入和输出必须是经典的呢？

Yuen originally went into computer science because he wanted to design video games.

Yuen 最初进入计算机科学领域是因为他想设计视频游戏。

“Looking back, I couldn’t have predicted any of the twists and turns that my interests have taken,” he said.

“回想起来，我无法预测我的兴趣所经历的任何曲折，”他说。

There’s a technique in cryptography called bit commitment, which is analogous to putting a message in a sealed envelope.

密码学中有一种称为比特承诺的技术，类似于将消息放入密封的信封中。

This preserves the privacy of the message until it’s time to reveal it, and it also prevents you from changing the message once it’s inside.

这可以保护消息的隐私，直到需要透露它为止，并且还可以防止您在消息进入后对其进行更改。

You could use a scheme like this to put in a bid for an auction or cast a secret vote.

您可以使用这样的方案来投标拍卖或进行无记名投票。

Classical bit commitment schemes are essential building blocks for all sorts of other methods in cryptography.

经典的比特承诺方案是密码学中各种其他方法的重要构建块。

But they all rely on the assumption that certain mathematical problems are intrinsically hard.

但它们都依赖于这样的假设：某些数学问题本质上是困难的。

If you somehow had the power to solve these famously hard problems, you could break any bit commitment scheme.

如果你有能力解决这些著名的难题，你就可以打破任何比特承诺方案。

Now imagine your envelopes are quantum.

现在想象你的信封是量子的。

In that case, even if you had that same amazing computational power, it’s not clear how you could use it to break these quantum envelopes.

在这种情况下，即使你拥有同样惊人的计算能力，也不清楚如何使用它来打破这些量子包络。

That’s right.

这是正确的。

The problem seems less mathematical and more physical.

这个问题似乎不那么数学化，而更物理化。

Somehow, having exorbitant classical computing power does not necessarily translate into exorbitant quantum processing power.

不知何故，拥有过高的经典计算能力并不一定会转化为过高的量子处理能力。

This is a hint that maybe these two worlds of computational tasks might be intrinsically different.

这暗示着这两个计算任务世界可能本质上是不同的。

In my opinion, the biggest open problem in this area is: What is the logical relationship between the two worlds?

在我看来，这个领域最大的开放性问题是：两个世界之间的逻辑关系是什么？

Yuen discusses a problem in quantum complexity theory with two graduate students.

袁与两名研究生讨论量子复杂性理论中的一个问题。

Suppose we knew everything there was to be known about traditional complexity theory.

假设我们知道有关传统复杂性理论的一切。

Would that tell us all the relationships between problems in fully quantum complexity theory?

这会告诉我们完全量子复杂性理论中问题之间的所有关系吗？

Maybe all of our understanding could just be easily ported over.

也许我们所有的理解都可以很容易地移植过来。

But it’s possible that fully quantum complexity theory is logically independent of traditional complexity.

但完全量子复杂性理论在逻辑上可能独立于传统复杂性。

This question has a beautiful formulation called the unitary synthesis problem .

这个问题有一个漂亮的表述，称为酉综合问题。

If I had access to an infinitely powerful classical oracle — a hypothetical device that could instantly solve any classical-input problem — could I use it to efficiently perform any quantum state transformation of my choice?

如果我能够使用一个无限强大的经典预言机（一种可以立即解决任何经典输入问题的假设设备），我可以用它来有效地执行我选择的任何量子态转换吗？

If the answer is no, that would mean these quantum-input, quantum-output problems are of an intrinsically different nature than classical ones.

如果答案是否定的，那就意味着这些量子输入、量子输出问题与经典问题有着本质上的不同。

There was a bit of progress on this problem a few years ago, but the full question is still wide open.

几年前，这个问题已经取得了一些进展，但完整的问题仍然悬而未决。

This is a project that my collaborators and I initiated a few years ago: To map out what this new world looks like, and how these quantum-input problems are related to each other.

这是我和我的合作者几年前发起的一个项目：绘制出这个新世界的样子，以及这些量子输入问题如何相互关联。

We found that one problem just showed up again and again in many different places.

我们发现一个问题在许多不同的地方一次又一次地出现。

In a recent paper, Yuen and his colleagues showed that many seemingly unrelated quantum problems are actually equivalent.

在最近的一篇论文中，袁和他的同事表明，许多看似无关的量子问题实际上是等价的。

It comes from a very fundamental result in quantum information theory called Uhlmann’s theorem.

它来自量子信息理论中一个非常基本的结果，称为乌尔曼定理。

Let’s say you have two quantum particles that are entangled, meaning they share a quantum state, and there are two possibilities for that shared state.

假设有两个纠缠的量子粒子，这意味着它们共享一个量子态，并且该共享态有两种可能性。

Quantum mechanics says it’s always possible to do some operations on both particles that transform the first state into the second state.

量子力学表示，总是可以对两个粒子进行一些操作，将第一个状态转变为第二个状态。

But what if I restrict you to only acting on one particle — can you still transform the first state into the second state?

但是，如果我限制你只对一个粒子起作用，你仍然可以将第一个状态转换为第二个状态吗？

If not, then what is the closest that you can get?

如果没有，那么你能得到的最接近的是什么？

Uhlmann’s theorem exactly quantifies the best that you can do for any pair of entangled states.

乌尔曼定理准确地量化了对任何一对纠缠态可以做的最好的事情。

It’s very important in quantum communication, quantum complexity, quantum cryptography, and beyond.

它在量子通信、量子复杂性、量子密码学等领域非常重要。

What we did was view the theorem as a quantum-input, quantum-output problem, where your goal is to transform one state into the other by only acting locally on one of the particles.

我们所做的是将定理视为量子输入、量子输出问题，其中您的目标是仅通过局部作用于其中一个粒子将一种状态转变为另一种状态。

So how hard is that problem?

那么这个问题有多难呢？

What resources do you need to solve it?

你需要什么资源来解决这个问题？

Is it equivalent in hardness to other problems that we care about?

它的难度与我们关心的其他问题相当吗？

We showed that there are actually a bunch of natural problems that are exactly equivalent in complexity.

我们证明，实际上存在许多复杂程度完全相同的自然问题。

One example is the computational problem of decoding the Hawking radiation of a black hole .

一个例子是解码黑洞霍金辐射的计算问题。

This is a quantum-input problem, because you’re scooping in and analyzing all these entangled particles.

这是一个量子输入问题，因为你正在挖掘并分析所有这些纠缠的粒子。

And it turns out that it’s just the Uhlmann transformation problem in disguise.

事实证明，这只是变相的乌尔曼变换问题。

The Uhlmann transformation is the hub from which all these other things radiate: bit commitments, black hole decoding, compressing quantum information, and more.

乌尔曼变换是所有其他事物辐射的中心：比特承诺、黑洞解码、压缩量子信息等等。

In 2020, Yuen helped prove a milestone result about the computational power of quantum entanglement.

2020 年，袁帮助证明了有关量子纠缠计算能力的里程碑式成果。

My parents grew up in Cambodia in the 1970s, when the Khmer Rouge took over the country and forced millions of people into labor camps.

我的父母在 20 世纪 70 年代在柬埔寨长大，当时红色高棉占领了这个国家，并迫使数百万人进入劳改营。

My mom’s family got swept into these camps, doing slave labor for three or four years, before they managed to escape.

我母亲的家人被卷入这些营地，做了三四年的奴隶劳动，然后才设法逃脱。

They spent months crossing mountains and land mine–filled fields.

他们花了几个月的时间穿越山脉和布满地雷的田野。

My dad’s family was somewhat more fortunate — they saw the warning signs and fled the country earlier.

我父亲的家人比较幸运——他们看到了警告信号并提前逃离了这个国家。

Eventually they managed to come to the U.S., and they ran a restaurant, where my brother and I worked until we left for school.

最终他们设法来到美国，并经营了一家餐馆，我和我的兄弟在那里工作，直到我们去上学。

Hearing their stories as a child obviously made a huge impression on me.

小时候听他们的故事显然给我留下了深刻的印象。

I get to enjoy this privilege of thinking about mathematics and quantum physics, and work with others who are interested in these extremely niche and insanely obscure topics.

我很享受思考数学和量子物理的特权，并与其他对这些极其利基和极其晦涩的主题感兴趣的人一起工作。

It’s so far removed from what they had to go through.

这与他们必须经历的事情相去甚远。

When I went to college, I planned to major in both physics and computer science — physics I just found conceptually interesting, computer science because of my high school interest in video game design.

当我上大学时，我计划主修物理和计算机科学——我只是发现物理在概念上很有趣，而计算机科学是因为我高中时对视频游戏设计感兴趣。

I quickly dropped physics because I was very bad at the labs.

我很快就放弃了物理学，因为我在实验室表现得很差。

Then in 2007 or 2008 I discovered Scott Aaronson’s blog , where he wrote about quantum computing and theoretical computer science.

然后在 2007 年或 2008 年，我发现了 Scott Aaronson 的博客，他在其中撰写了有关量子计算和理论计算机科学的文章。

I was like, “This is amazing.

我当时想，“这太棒了。

I want to study this.” That planted the seed, but the biggest formative experience for me was doing undergraduate research with Len Adleman .

我想研究这个。”这埋下了种子，但对我来说影响最大的经历是和 Len Adleman 一起进行本科研究。

His mantra was: “We’re going to ignore everything that people have done for 100 years.

他的口头禅是：“我们将忽视人们一百年来所做的一切。

We’re going to take a left turn where everyone took a right turn.” It felt like we were mavericks, doing something a bit rebellious.

我们要在所有人都右转的地方左转。”感觉就像我们是特立独行的人，做了一些有点叛逆的事情。

It’s a very different exercise from what I’m used to.

这是与我习惯的非常不同的练习。

No one hands you a theorem and says, “Here, prove it.” We don’t even necessarily know what the right questions are.

没有人递给你一个定理并说：“来证明一下。”我们甚至不一定知道什么是正确的问题。

But finding the right language is really important, even if you don’t prove anything really technical.

但找到正确的语言确实很重要，即使你没有证明任何真正的技术性东西。

Not having the right language actually prevents you from thinking clearly.

没有正确的语言实际上会妨碍你清晰地思考。

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获取发送到您的电子邮件收件箱的最重要新闻的摘要

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《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。

Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.

版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。